Geometric Sequence – Sum of Last Terms | Grade 11

Geometric Sequence

Sum of the Last Terms

Sometimes we need the sum of the last k terms in the sequence.

For example: from a sequence of 10 terms, compute the sum of the last 3.

For the sum of the last k terms there is no special formula

We will use other calculation methods

Sum of the last k terms = \(S_n - S_{n-k}\)

💡 Explanation:

Sₙ = sum of all n terms

Sₙ₋ₖ = sum of the first (n−k) terms

The difference = the last k terms

✏️ Example: Geometric sequence a₁ = 2, q = 3, n = 8. Compute the sum of the last 3 terms.

Solution:

Sum of last 3 terms = S₈ − S₅

\(S_8 = \frac{2(3^8 - 1)}{3-1} = \frac{2 \cdot 6560}{2} = 6560\)

\(S_5 = \frac{2(3^5 - 1)}{3-1} = \frac{2 \cdot 242}{2} = 242\)

Sum of last 3 terms = 6560 − 242 = 6318

💡 Idea: The last k terms form their own geometric sequence with the same ratio q!

Sum of the last k terms = \(\frac{a_{n-k+1}(q^k - 1)}{q - 1}\)

✏️ Same example: a₁ = 2, q = 3, n = 8. Sum of the last 3 terms.

Solution:

First term of the last 3: a₆ = a₈₋₃₊₁

\(a_6 = 2 \cdot 3^5 = 486\)

Sum: \(\frac{486(3^3 - 1)}{3-1} = \frac{486 \cdot 26}{2} = 6318\)

Same answer! ✓

Sum of the last k terms = \(S_k \cdot q^{n-k}\)

✏️ Same example:

Solution:

\(S_3 = \frac{2(3^3 - 1)}{3-1} = \frac{2 \cdot 26}{2} = 26\)

Sum of last 3 terms: \(26 \cdot 3^{8-3} = 26 \cdot 243 = 6318\)

Same answer! ✓

Method	Formula	When convenient
Subtracting sums	\(S_n - S_{n-k}\)	When a sum formula is available
New sequence	\(\frac{a_{n-k+1}(q^k-1)}{q-1}\)	When one of the last terms is given
Relationship between sums	\(S_k \cdot q^{n-k}\)	When Sₖ is easy to compute

There is no direct formula for the sum of the last terms

Convenient method: Sₙ − Sₙ₋ₖ = sum of the last k terms

Or: Sₖ · qⁿ⁻ᵏ