Geometric Sequence — Sum of Last Terms and General Term from Sum Formula

Geometric Sequence — Sum of Last Terms and General Term from Sum Formula. Practice questions to deepen understanding of advanced topics in geometric sequences. Online math practice with full solutions and step-by-step explanations.

Advanced geometric sequence practice — sum of last terms, general term from the sum formula, convergent infinite sequences. Advanced level.

80 questions

Question 1
2.50 pts

📚 :

" 5 "?

Explanation:

💡 :

! 📚

: a₁, a₂, a₃, ..., aₙ₋₄, aₙ₋₃, aₙ₋₂, aₙ₋₁, aₙ
Question 2
10.00 pts

📚 Understanding:

What is meant by "the sum of the last 5 terms"?

Explanation:
NULL
Question 3
10.00 pts

❓ Conceptual question:

Why is there no direct formula for the sum of the last k terms?

Explanation:
NULL
Question 4
2.50 pts

❓ Conceptual question:

Why is there no direct formula for the sum of the last terms?

Explanation:

💡 Detailed explanation:

Why no direct formula? ❓

The existing sum formula:

Sₙ = a₁(qⁿ-1)/(q-1)

The problem:
This formula computes the sum from the beginning (from a₁)

There is no direct formula starting from the middle!

Solution:
We need to compute:
Total sum - sum of first terms = sum of last terms

⭐ This is the subtraction method!
Question 5
2.50 pts

➖ elimination method:

3 10 ?

Explanation:

💡 :

elimination method! ➖

10 : a₁, a₂, ..., a₇, a₈, a₉, a₁₀
Question 6
10.00 pts

➖ Subtraction method:

How do you find the sum of the last 3 terms of a sequence of 10 terms?

Explanation:
NULL
Question 7
10.00 pts

🔢 Calculation:

Sequence: 2, 6, 18, 54, 162.

What is the sum of the last 2 terms?

Explanation:
NULL
Question 8
2.50 pts

🔢 Computation:

Sequence: 2, 6, 18, 54, 162

What is the sum of the last 2 terms?

Explanation:

💡 Detailed explanation:

Computing the last 2 terms! 🔢

The sequence:
a₁=2, a₂=6, a₃=18, a₄=54, a₅=162

The last 2 terms:
a₄ + a₅ = 54 + 162 = 216

Or by subtraction method:
S₅ = 2+6+18+54+162 = 242
S₃ = 2+6+18 = 26
S₅ - S₃ = 242 - 26 = 216 ✓

⭐ Both methods give the same result!
Question 9
2.50 pts

🔢 Using the formula:

Geometric sequence: a₁=3, q=2, n=8

What is the sum of the last 3 terms?

Explanation:

💡 Detailed explanation:

Using the sum formula! 🔢

Given:
a₁ = 3, q = 2, n = 8

Need: sum of last 3 = S₈ - S₅

Compute S₈:
S₈ = 3(2⁸-1)/(2-1) = 3×255 = 765

Compute S₅:
S₅ = 3(2⁵-1)/(2-1) = 3×31 = 93

Sum of last 3:
S₈ - S₅ = 765 - 93 = 576

⭐ This equals a₆ + a₇ + a₈
Question 10
10.00 pts

🔢 Using the formula:

Geometric sequence: a₁ = 3, q = 2, n = 8.

What is the sum of the last 3 terms?

Explanation:
NULL
Question 11
10.00 pts

⚠️ Common error:

A student wants to find the sum of the last 4 terms of a 12-term sequence.
The student writes S₁₂ − S₄.

What is wrong?

Explanation:
NULL
Question 12
2.50 pts

⚠️ Common mistake:

A student wants to find the sum of the last 4 terms in a 12-term sequence.
He computed S₁₂ - S₄

What is the mistake?

Explanation:

💡 Detailed explanation:

The common mistake! ⚠️

❌ Wrong calculation:
S₁₂ - S₄

What does this give?
It gives the sum of terms from a₅ to a₁₂
That is 8 terms! (not 4)

✅ Correct calculation:
Last 4 in a 12-term sequence:
Need from a₉ to a₁₂

S₁₂ - S₈ = sum of last 4

⭐ Rule:
Last k terms = Sₙ - Sₙ₋ₖ
Last 4 of 12 = S₁₂ - S₈
Question 13
2.50 pts

📐 :

k n ?

Explanation:

💡 :

! 📐

k n
Question 14
10.00 pts

📐 General formula:

How do you find the sum of the last k terms of a sequence of n terms?

Explanation:
NULL
Question 15
10.00 pts

🔢 Exercise:

a₁ = 1, q = 3, n = 10.

What is the sum of the last 5 terms?

Explanation:
NULL
Question 16
2.50 pts

🔢 Exercise:

a₁=1, q=3, n=10

Sum of the last 5 terms?

Explanation:

💡 Detailed explanation:

Computing last 5 terms! 🔢

Need: S₁₀ - S₅

Compute S₁₀:
S₁₀ = 1×(3¹⁰-1)/(3-1)
S₁₀ = (59,049-1)/2
S₁₀ = 59,048/2 = 29,524

Compute S₅:
S₅ = 1×(3⁵-1)/(3-1)
S₅ = (243-1)/2
S₅ = 242/2 = 121

Result:
29,524 - 121 = 14,640

This is a₆+a₇+a₈+a₉+a₁₀
Question 17
2.50 pts

🔢 :

a₁=2, q=5, n=6

" " (1 )?

Explanation:

💡 :

! 🔢

1:
a₆ = a₁ × q⁵
a₆ = 2 × 5⁵
a₆ = 2 × 3,125
a₆ = 6,250

2: elimination method
S₆ - S₅
S₆ = 2(5⁶-1)/4 = 2×3,906 = 7,812
S₅ = 2(5⁵-1)/4 = 2×781 = 1,562
7,812 - 1,562 = 6,250 ✓

⭐ No !
Question 18
10.00 pts

🔢 Special case:

a₁ = 2, q = 5, n = 6.

What is the "sum of the last term only" (1 term)?

Explanation:
NULL
Question 19
10.00 pts

🔢 Exercise:

a₁ = 5, q = 2, n = 7.

What is the sum of the last 2 terms?

Explanation:
NULL
Question 20
2.50 pts

🔢 Exercise:

a₁=5, q=2, n=7

Sum of the last 2 terms?

Explanation:

💡 Detailed explanation:

Last 2 terms! 🔢

Need: S₇ - S₅

Compute S₇:
S₇ = 5(2⁷-1)/(2-1)
S₇ = 5×127 = 635

Compute S₅:
S₅ = 5(2⁵-1)/(2-1)
S₅ = 5×31 = 155

Result:
635 - 155 = 480

Direct check:
a₆ = 5×2⁵ = 160
a₇ = 5×2⁶ = 320
160 + 320 = 480 ✓
Question 21
2.50 pts

🔢 Exercise:

a₁=4, q=3, n=9

Sum of the last 4 terms?

Explanation:

💡 Detailed explanation:

Last 4 terms! 🔢

Need: S₉ - S₅

Compute S₉:
S₉ = 4(3⁹-1)/(3-1)
S₉ = 4×(19,683-1)/2
S₉ = 4×9,841 = 39,364

Compute S₅:
S₅ = 4(3⁵-1)/(3-1)
S₅ = 4×(243-1)/2
S₅ = 4×121 = 484

Last 4:
S₉ - S₅ = 39,364 - 484 = 26,244

This is a₆ + a₇ + a₈ + a₉
Question 22
10.00 pts

🔢 Exercise:

a₁ = 4, q = 3, n = 9.

What is the sum of the last 4 terms?

Explanation:
NULL
Question 23
10.00 pts

🔢 Decimal ratio:

a₁ = 100, q = 0.5, n = 8.

What is the sum of the last 3 terms?

Explanation:
NULL
Question 24
2.50 pts

🔢 Decimal ratio:

a₁=100, q=0.5, n=8

Sum of the last 3 terms?

Explanation:

💡 Detailed explanation:

Ratio less than 1! 🔢

Need: S₈ - S₅

Compute S₈:
S₈ = 100(1-0.5⁸)/(1-0.5)
S₈ = 100×(1-0.00390625)/0.5
S₈ = 100×1.9921875 = 199.21875

Compute S₅:
S₅ = 100(1-0.5⁵)/(1-0.5)
S₅ = 100×(1-0.03125)/0.5
S₅ = 100×1.9375 = 193.75

Last 3:
199.21875 - 193.75 = 4.6875

⭐ When q<1 the sequence decreases!
Question 25
2.50 pts

🔍 :

a₁=2, q=3, n=8
4,920

?

Explanation:

💡 :

! 🔍

given:
S₈ - Sₖ = 4,920

S₈:
S₈ = 2(3⁸-1)/(3-1)
S₈ = 2×3,280 = 6,560

Yes:
6,560 - Sₖ = 4,920
Sₖ = 6,560 - 4,920 = 1,640

k:
1,640 = 2(3ᵏ-1)/2
1,640 = 3ᵏ - 1
3ᵏ = 1,641
: 3⁴ = 81 ✗, 3⁵ = 243 ✗
! : 3⁴ = 81, Yes...
: k=4, because S₄ = 2×40 = 80.
Question 26
10.00 pts

🔍 Reverse problem:

a₁ = 2, q = 3, n = 8.
The sum of the last k terms is 4,920.

How many terms are included?

Explanation:
NULL
Question 27
10.00 pts

📊 Half the sequence:

In a sequence of 10 terms: a₁ = 3, q = 2.

What is the sum of the last 5 terms?

Explanation:
NULL
Question 28
2.50 pts

📊 Half the sequence:

In a 10-term sequence: a₁=3, q=2

Sum of the last 5 terms?

Explanation:

💡 Detailed explanation:

Last half! 📊

Need: S₁₀ - S₅

Compute S₁₀:
S₁₀ = 3(2¹⁰-1)/(2-1)
S₁₀ = 3×1,023 = 3,069

Compute S₅:
S₅ = 3(2⁵-1)/(2-1)
S₅ = 3×31 = 93

Last half:
3,069 - 93 = 3,024

⭐ The last half is much larger!
(because q>1)
Question 29
2.50 pts

⚖️ :

: a₁=1, q=2, n=10

large : 5 5 ?

Explanation:

💡 :

! ⚖️

5
Question 30
10.00 pts

⚖️ Comparison:

Sequence: a₁ = 1, q = 2, n = 10.

Which is larger: the sum of the first 5 or the last 5 terms?

Explanation:
NULL
Question 31
10.00 pts

🔢 Exercise:

a₁ = 1, q = 4, n = 8.

What is the sum of the last 6 terms?

Explanation:
NULL
Question 32
2.50 pts

🔢 Exercise:

a₁=1, q=4, n=8

Sum of the last 6 terms?

Explanation:

💡 Detailed explanation:

Last 6 terms! 🔢

Need: S₈ - S₂

Compute S₈:
S₈ = 1×(4⁸-1)/(4-1)
S₈ = (65,536-1)/3
S₈ = 65,535/3 = 21,845

Compute S₂:
S₂ = 1×(4²-1)/(4-1)
S₂ = 15/3 = 5

Last 6:
21,845 - 5 = 21,840

This is a₃ + a₄ + a₅ + a₆ + a₇ + a₈
Question 33
2.50 pts

🤔 :

15 , S₁₅ - S₁₀

?

Explanation:

💡 :

! 🤔

a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈, a₉, a₁₀, a₁₁, a₁₂, a₁₃, a₁₄, a₁₅
Question 34
10.00 pts

🤔 Understanding:

In a sequence of 15 terms, we calculated S₁₅ − S₁₀.

What does this represent?

Explanation:
NULL
Question 35
10.00 pts

🔢 Exercise:

a₁ = 2, q = 2, n = 12.

What is the sum of the last 7 terms?

Explanation:
NULL
Question 36
2.50 pts

🔢 Exercise:

a₁=2, q=2, n=12

Sum of the last 7 terms?

Explanation:

💡 Detailed explanation:

Last 7 terms! 🔢

Need: S₁₂ - S₅

Compute S₁₂:
S₁₂ = 2(2¹²-1)/(2-1)
S₁₂ = 2×4,095 = 8,190

Compute S₅:
S₅ = 2(2⁵-1)/(2-1)
S₅ = 2×31 = 62

Last 7:
8,190 - 62 = 8,128

This is a₆ + a₇ + a₈ + a₉ + a₁₀ + a₁₁ + a₁₂
Question 37
2.50 pts

💰 Application:

You deposited $1,000 in the bank. Each year the money is multiplied by 1.1
After 10 years, how much did you earn in the last 3 years?

Explanation:

💡 Detailed explanation:

Earnings in the last years! 💰

Setup:
a₁ = 1,000
q = 1.1
n = 10

Need: sum of last 3 = S₁₀ - S₇

Compute S₁₀:
S₁₀ = 1,000(1.1¹⁰-1)/(1.1-1)
S₁₀ = 1,000×(2.5937-1)/0.1
S₁₀ ≈ 15,937

Compute S₇:
S₇ = 1,000(1.1⁷-1)/0.1
S₇ ≈ 9,487

Last 3:
15,937 - 9,487 ≈ 6,450

Note: the actual gain is the difference from principal, so a different calculation is needed.
Question 38
10.00 pts

💰 Application:

1,000 was deposited in a bank. Each year the amount is multiplied by 1.1.
After 10 years, what is the sum of the last 3 years?

Explanation:
NULL
Question 39
10.00 pts

📋 Summary:

What is the core principle for finding the sum of the last terms?

Explanation:
NULL
Question 40
2.50 pts

📋 :

?

Explanation:

💡 :

! 📋

Question 41
2.50 pts

📐 :

aₙ Sₙ?

Explanation:

💡 :

! 📐

a₁ + a₂ + a₃ + ... + aₙ₋₁ + aₙ
Question 42
10.00 pts

📐 Principle:

How do you find aₙ from the formula Sₙ?

Explanation:
NULL
Question 43
10.00 pts

❓ Question:

Why does the formula aₙ = Sₙ − Sₙ₋₁ work only for n ≥ 2?

Explanation:
NULL
Question 44
2.50 pts

❓ Question:

Why does the formula aₙ = Sₙ - Sₙ₋₁ work only for n≥2?

Explanation:

💡 Detailed explanation:

Why only n≥2? ❓

What happens when n=1?

By the formula:
a₁ = S₁ - S₀

The problem:
S₀ = sum of 0 terms?
This is not defined!

The solution:
a₁ has a special rule:
a₁ = S₁

For all the rest (n≥2):
aₙ = Sₙ - Sₙ₋₁

⭐ Two rules: one for a₁, one for the rest!
Question 45
2.50 pts

🔢 Example:

Given: Sₙ = 3n² + 2n

What is a₁?

Explanation:

💡 Detailed explanation:

Finding a₁! 🔢

Given:
Sₙ = 3n² + 2n

Special rule for a₁:
a₁ = S₁

Computation:
a₁ = S₁ = 3×1² + 2×1
a₁ = 3×1 + 2
a₁ = 3 + 2
a₁ = 5

Check:
S₁ should equal a₁ alone
S₁ = 5 ✓
Question 46
10.00 pts

🔢 Example:

Given: Sₙ = 3n² + 2n.

What is a₁?

Explanation:
NULL
Question 47
10.00 pts

🔢 Example:

Given: Sₙ = 3n² + 2n.

What is a₂?

Explanation:
NULL
Question 48
2.50 pts

🔢 Example:

Given: Sₙ = 3n² + 2n

What is a₂?

Explanation:

💡 Detailed explanation:

Finding a₂! 🔢

Given:
Sₙ = 3n² + 2n

Formula:
a₂ = S₂ - S₁

Compute S₂:
S₂ = 3×2² + 2×2
S₂ = 3×4 + 4
S₂ = 12 + 4 = 16

Compute S₁:
S₁ = 3×1² + 2×1 = 5
(this is a₁ found earlier)

Therefore:
a₂ = 16 - 5 = 11

Check: a₁ + a₂ = 5 + 11 = 16 = S₂ ✓
Question 49
2.50 pts

🔢 Example:

Given: Sₙ = 3n² + 2n

What is a₃?

Explanation:

💡 Detailed explanation:

Finding a₃! 🔢

Formula:
a₃ = S₃ - S₂

Compute S₃:
S₃ = 3×3² + 2×3
S₃ = 3×9 + 6
S₃ = 27 + 6 = 33

Compute S₂:
S₂ = 16 (computed earlier)

Therefore:
a₃ = 33 - 16 = 17

The sequence so far:
a₁ = 5, a₂ = 11, a₃ = 17

⭐ See the pattern? It is an arithmetic sequence! d=6
Question 50
10.00 pts

🔢 Example:

Given: Sₙ = 3n² + 2n.

What is a₃?

Explanation:
NULL
Question 51
10.00 pts

📐 General formula:

Given: Sₙ = 3n² + 2n.

What is the general formula for aₙ (for n ≥ 2)?

Explanation:
NULL
Question 52
2.50 pts

📐 General formula:

Given: Sₙ = 3n² + 2n

What is the general term formula aₙ (for n≥2)?

Explanation:

💡 Detailed explanation:

Deriving the term formula! 📐

Given:
Sₙ = 3n² + 2n

Computation:
aₙ = Sₙ - Sₙ₋₁

Sₙ = 3n² + 2n
Sₙ₋₁ = 3(n-1)² + 2(n-1)
Sₙ₋₁ = 3(n² - 2n + 1) + 2n - 2
Sₙ₋₁ = 3n² - 6n + 3 + 2n - 2
Sₙ₋₁ = 3n² - 4n + 1

Subtraction:
aₙ = (3n² + 2n) - (3n² - 4n + 1)
aₙ = 3n² + 2n - 3n² + 4n - 1
aₙ = 6n - 1

Check:
a₂ = 6×2 - 1 = 11 ✓
a₃ = 6×3 - 1 = 17 ✓
Question 53
2.50 pts

✅ Verification:

We found: aₙ = 6n - 1
a₁ = 5

Does the formula also work for a₁?

Explanation:

💡 Detailed explanation:

Checking the formula! ✅

The formula we found:
aₙ = 6n - 1 (for n≥2)

Check for n=1:
a₁ = 6×1 - 1 = 5

What we computed from S₁:
a₁ = S₁ = 5

Result:
5 = 5 ✓

Conclusion:
In this case the formula also works for a₁!

So we can write:
aₙ = 6n - 1 for all n≥1

⭐ This does not always happen - always check!
Question 54
10.00 pts

✅ Check:

We found: aₙ = 6n − 1, a₁ = 5.

Does the formula work for a₁?

Explanation:
NULL
Question 55
10.00 pts

🔢 New example:

Given: Sₙ = 2ⁿ − 1.

What is a₁?

Explanation:
NULL
Question 56
2.50 pts

🔢 New example:

Given: Sₙ = 2ⁿ - 1

What is a₁?

Explanation:

💡 Detailed explanation:

Finding a₁! 🔢

Given:
Sₙ = 2ⁿ - 1

Rule:
a₁ = S₁

Computation:
a₁ = S₁ = 2¹ - 1
a₁ = 2 - 1
a₁ = 1

Insight:
This is the sum formula of a geometric sequence!
a₁=1, q=2

⭐ Recognize the formula: Sₙ = a₁(qⁿ-1)/(q-1)
Question 57
2.50 pts

🔢 Continued:

Given: Sₙ = 2ⁿ - 1

What is a₂?

Explanation:

💡 Detailed explanation:

Finding a₂! 🔢

Formula:
a₂ = S₂ - S₁

Compute S₂:
S₂ = 2² - 1
S₂ = 4 - 1 = 3

Compute S₁:
S₁ = 1 (found earlier)

Therefore:
a₂ = 3 - 1 = 2

The sequence:
a₁ = 1, a₂ = 2

The ratio: q = a₂/a₁ = 2/1 = 2 ✓

⭐ This is a geometric sequence with q=2!
Question 58
10.00 pts

🔢 Continued:

Given: Sₙ = 2ⁿ − 1.

What is a₂?

Explanation:
NULL
Question 59
10.00 pts

📐 General formula:

Given: Sₙ = 2ⁿ − 1.

What is the general formula for aₙ?

Explanation:
NULL
Question 60
2.50 pts

📐 General formula:

Given: Sₙ = 2ⁿ - 1

What is the general term formula aₙ?

Explanation:

💡 Detailed explanation:

Deriving the formula! 📐

Computation:
aₙ = Sₙ - Sₙ₋₁

Sₙ = 2ⁿ - 1
Sₙ₋₁ = 2ⁿ⁻¹ - 1

Subtraction:
aₙ = (2ⁿ - 1) - (2ⁿ⁻¹ - 1)
aₙ = 2ⁿ - 1 - 2ⁿ⁻¹ + 1
aₙ = 2ⁿ - 2ⁿ⁻¹
aₙ = 2ⁿ⁻¹(2 - 1)
aₙ = 2ⁿ⁻¹

Check:
a₁ = 2⁰ = 1 ✓
a₂ = 2¹ = 2 ✓
a₃ = 2² = 4

⭐ This is exactly the geometric sequence formula: aₙ = a₁ × qⁿ⁻¹
Question 61
2.50 pts

🔢 New example:

Given: Sₙ = n² + n

What is a₁?

Explanation:

💡 Detailed explanation:

Finding a₁! 🔢

Given:
Sₙ = n² + n

Computation:
a₁ = S₁
a₁ = 1² + 1
a₁ = 1 + 1
a₁ = 2

We can factor:
Sₙ = n² + n = n(n + 1)

Therefore: S₁ = 1(1 + 1) = 2 ✓
Question 62
10.00 pts

🔢 New example:

Given: Sₙ = n² + n.

What is a₁?

Explanation:
NULL
Question 63
10.00 pts

📐 Formula:

Given: Sₙ = n² + n.

What is the formula for aₙ (for n ≥ 2)?

Explanation:
NULL
Question 64
2.50 pts

📐 Formula:

Given: Sₙ = n² + n

What is the general term formula aₙ (for n≥2)?

Explanation:

💡 Detailed explanation:

Deriving the formula! 📐

Computation:
aₙ = Sₙ - Sₙ₋₁

Sₙ = n² + n
Sₙ₋₁ = (n-1)² + (n-1)
Sₙ₋₁ = n² - 2n + 1 + n - 1
Sₙ₋₁ = n² - n

Subtraction:
aₙ = (n² + n) - (n² - n)
aₙ = n² + n - n² + n
aₙ = 2n

Check:
a₂ = 2×2 = 4
S₂ = 2² + 2 = 6 = a₁ + a₂ = 2 + 4 ✓

⭐ This is an arithmetic sequence! d=2
Question 65
2.50 pts

⚠️ Verification:

We found: aₙ = 2n, a₁ = 2

Does the formula work for a₁?

Explanation:

💡 Detailed explanation:

Verification! ⚠️

The formula: aₙ = 2n

Check for n=1:
a₁ = 2×1 = 2

Direct computation from S₁:
a₁ = S₁ = 1² + 1 = 2

Result:
2 = 2 ✓

Conclusion:
In this case the formula also works for a₁!

So: aₙ = 2n for all n≥1

The sequence: 2, 4, 6, 8, 10...

⭐ Even numbers!
Question 66
10.00 pts

⚠️ Check:

We found: aₙ = 2n, a₁ = 2.

Does the formula work for a₁?

Explanation:
NULL
Question 67
10.00 pts

🔢 Example:

Given: Sₙ = n² + 2n + 1.

What is a₁?

Explanation:
NULL
Question 68
2.50 pts

🔢 Example:

Given: Sₙ = n² + 2n + 1

What is a₁?

Explanation:

💡 Detailed explanation:

Finding a₁! 🔢

Given:
Sₙ = n² + 2n + 1

Computation:
a₁ = S₁
a₁ = 1² + 2×1 + 1
a₁ = 1 + 2 + 1
a₁ = 4

Note:
We can see that:
Sₙ = n² + 2n + 1 = (n + 1)²

Therefore: S₁ = (1 + 1)² = 2² = 4 ✓
Question 69
2.50 pts

📐 Formula:

Given: Sₙ = n² + 2n + 1

What is the formula for aₙ (for n≥2)?

Explanation:

💡 Detailed explanation:

Derivation! 📐

Computation:
Sₙ = n² + 2n + 1
Sₙ₋₁ = (n-1)² + 2(n-1) + 1
Sₙ₋₁ = n² - 2n + 1 + 2n - 2 + 1
Sₙ₋₁ = n² + 0n + 0 = n²

Subtraction:
aₙ = (n² + 2n + 1) - n²
aₙ = 2n + 1

Check for a₁:
a₁ = 2×1 + 1 = 3
But we computed a₁ = 4 ✗

Conclusion:
The formula aₙ = 2n + 1 works only for n≥2
a₁ has a special value: a₁ = 4

⭐ The sequence: 4, 5, 7, 9, 11...
Question 70
10.00 pts

📐 Formula:

Given: Sₙ = n² + 2n + 1.

What is the formula for aₙ (for n ≥ 2)?

Explanation:
NULL
Question 71
10.00 pts

🔢 Application:

Given: Sₙ = 2n² − n.

What is a₅?

Explanation:
NULL
Question 72
2.50 pts

🔢 [MATH_FLAG] Application:

Given: Sₙ = 2n² - n

What is a₅?

Note: source contains a calculation error in the explanation; please verify the marked answer.

Explanation:

💡 Detailed explanation:

Finding a₅! [MATH_FLAG] 🔢

Formula:
a₅ = S₅ - S₄

Compute S₅:
S₅ = 2×5² - 5
S₅ = 2×25 - 5
S₅ = 50 - 5 = 45

Compute S₄:
S₄ = 2×4² - 4
S₄ = 2×16 - 4
S₄ = 32 - 4 = 28

Therefore:
a₅ = 45 - 30 = 15

⚠ Note: there is a calculation discrepancy. The arithmetic 45 - 28 = 17, not 15. The source explanation acknowledges: "Wait, calculation error: a₅ = 45 - 28 = 17... oops! Need to fix the answer." Please review and correct.
Question 73
2.50 pts

📐 Exercise:

Given: Sₙ = n³

What is the formula for aₙ (for n≥2)?

Explanation:

💡 Detailed explanation:

Derivation! 📐

Computation:
Sₙ = n³
Sₙ₋₁ = (n-1)³

Expand:
(n-1)³ = n³ - 3n² + 3n - 1

Subtraction:
aₙ = n³ - (n³ - 3n² + 3n - 1)
aₙ = n³ - n³ + 3n² - 3n + 1
aₙ = 3n² - 3n + 1

Check:
a₁ = S₁ = 1³ = 1
By the formula: a₁ = 3×1 - 3×1 + 1 = 1 ✓

a₂ = S₂ - S₁ = 8 - 1 = 7
By the formula: a₂ = 3×4 - 6 + 1 = 7 ✓

⭐ The formula works for all n≥1!
Question 74
10.00 pts

📐 Exercise:

Given: Sₙ = n³.

What is the formula for aₙ (for n ≥ 2)?

Explanation:
NULL
Question 75
10.00 pts

🔍 Identification:

Given: Sₙ = 5n.

What kind of sequence is this?

Explanation:
NULL
Question 76
2.50 pts

🔍 Identification:

Given: Sₙ = 5n

What kind of sequence is this?

Explanation:

💡 Detailed explanation:

Identifying the sequence! 🔍

Given: Sₙ = 5n

Finding a₁:
a₁ = S₁ = 5×1 = 5

Finding aₙ:
aₙ = Sₙ - Sₙ₋₁
aₙ = 5n - 5(n-1)
aₙ = 5n - 5n + 5
aₙ = 5

The sequence:
5, 5, 5, 5, 5, 5...

Conclusion:
This is a constant sequence!

All terms equal 5

⭐ Can be viewed as:
• Arithmetic with d=0
• Geometric with q=1
Question 77
2.50 pts

🔢 Exercise:

Given: Sₙ = 3ⁿ - 2

What is a₃?

Explanation:

💡 Detailed explanation:

Computing a₃! 🔢

Formula:
a₃ = S₃ - S₂

Compute S₃:
S₃ = 3³ - 2
S₃ = 27 - 2 = 25

Compute S₂:
S₂ = 3² - 2
S₂ = 9 - 2 = 7

Therefore:
a₃ = 25 - 7 = 18

The sequence:
a₁ = S₁ = 3 - 2 = 1
a₂ = S₂ - S₁ = 7 - 1 = 6
a₃ = 18

The ratio: 6/1 = 6, 18/6 = 3... not geometric!

⭐ But aₙ = 3ⁿ - 3ⁿ⁻¹ = 3ⁿ⁻¹(3-1) = 2×3ⁿ⁻¹
Question 78
10.00 pts

🔢 Exercise:

Given: Sₙ = 3ⁿ − 2.

What is a₃?

Explanation:
NULL
Question 79
10.00 pts

📋 Summary:

What are the steps for finding aₙ from Sₙ?

Explanation:
NULL
Question 80
2.50 pts

📋 :

aₙ -Sₙ?

Explanation:

💡 :

! 📋

aₙ -Sₙ← Back to all exams