Let $S _{ n }=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots$ 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
- [JEE MAIN 2025]
- A$25$
- B$90$
- C$20$
- D$45$
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The number of terms of an $A.P.$ is even; the sum of all the odd terms is $24$ , the sum of all the even terms is $30$ and the last term exceeds the first by $\frac{21}{2}$. Then the number of terms which are integers in the $A.P.$ is :
- [JEE MAIN 2025]
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- [IIT 2019]
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Let the sequence $a_{n}$ be defined as follows:
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Find first five terms and write corresponding series.
Let the sequence $a_{n}$ be defined as follows:
${a_1} = 1,{a_n} = {a_{n - 1}} + 2$ for $n\, \ge \,2$
Find first five terms and write corresponding series.