Sequences & Series
GP with Each Term as AM — S₂₀−S₁₈
nta_pyq_2024_jan
Grade 11
Question:
If each term of a geometric progression $a_1,a_2,a_3,\ldots$ with $a_1=\dfrac{1}{8}$ and $a_2\neq a_1$, is the arithmetic mean of the next two terms and $S_n=a_1+a_2+\cdots+a_n$, then $S_{20}-S_{18}$ is equal to
$2^{15}$
$-2^{18}$
$2^{18}$
$-2^{15}$
Step-by-Step Solution
Key Concept: $2a_n=a_{n+1}+a_{n+2}$ (each term is AM of next two). With $a_n=ar^{n-1}$: $2=r+r^2\Rightarrow r^2+r-2=0\Rightarrow r=-2$ (since $r\neq1$). $S_{20}-S_{18}=T_{19}+T_{20}=ar^{18}(1+r)$.
$r=-2$. $S_{20}-S_{18}=ar^{18}(1+r)=\frac{1}{8}\cdot2^{18}\cdot(-1)=-2^{15}$.
Correct Answer: 4