Sequences & Series
Sum of AP with Ratio Condition
nta_pyq_2025_apr
Grade 11
Question:
Let the first term of an AP be 3. If the sum of its first 4 terms is $\dfrac{1}{5}$ of the sum of the next 4 terms, then $S_{20}$ equals
$-1080$
$-1020$
$-1200$
$-120$
Step-by-Step Solution
Key Concept: Express the condition $5S_4 = S_8-S_4$, i.e. $6S_4=S_8$, in terms of $a=3$ and $d$, then solve for $d$.
$S_4=\frac{1}{5}(S_8-S_4)\Rightarrow 6S_4=S_8$.
$S_4=4(3)+6d=12+6d$; $S_8=8(3)+28d=24+28d$.
$6(12+6d)=24+28d\Rightarrow72+36d=24+28d\Rightarrow8d=-48\Rightarrow d=-6$.
$S_{20}=10\bigl(2(3)+19(-6)\bigr)=10(6-114)=10(-108)=-1080$.
Correct Answer: 1