In an arithmetic progression, the sum of the first 8 terms is 200 and the sum of the first 4 terms is 80. Find the 8th term.

Step-by-step solution

1. Let the first term be $a$ and the common difference be $d$.
2. Use $S_n=\frac{n}{2}[2a+(n-1)d]$.
3. For $n=8$: $S_8=4[2a+7d]=200\Rightarrow 2a+7d=50$ ...(1).
4. For $n=4$: $S_4=2[2a+3d]=80\Rightarrow 2a+3d=40$ ...(2).
5. Subtract (2) from (1): $(2a+7d)-(2a+3d)=4d=10\Rightarrow d=\frac{10}{4}=2.5$.
6. Substitute $d$ into (2): $2a+3(2.5)=40\Rightarrow 2a+7.5=40\Rightarrow 2a=32.5\Rightarrow a=16.25$.
7. The 8th term is $a+7d=16.25+7\times 2.5=16.25+17.5=33.75$.

Concept tested: using two sums to find a term

Published 25 August 2026 · How this solution was prepared and checked

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