Let $S_n=\frac12+\frac16+\frac1{12}+\frac1{20}+\dots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026\,S_{2025}}$, then the absolute difference between $20^{ ext{th}}$ and $15^{ ext{th}}$ terms of the A.P. is

Question

Let $S_n=\frac12+\frac16+\frac1{12}+\frac1{20}+\dots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026\,S_{2025}}$, then the absolute difference between $20^{	ext{th}}$ and $15^{	ext{th}}$ terms of the A.P. is

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] $S_n=\sum\frac1{k(k+1)}=\frac n{n+1}$, so $2026S_{2025}=2025$ and $\sqrt{2026S_{2025}}=45$. Sum of first six terms of the A.P. $=\frac62(-2p+5p)=9p=45\Rightarrow p=5$. Difference of the $20^{	ext{th}}$ and $15^{	ext{th}}$ terms $=5p=25$. Answer (4).

Q. Let $S_n=\frac12+\frac16+\frac1{12}+\frac1{20}+\dots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026\,S_{2025}}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is

Explanation

Reviewed by Dr. Abubacker

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