If sum of series 54 + 51 + 48 + ............. is 513 , then number of terms are

English

Gujarati

Hindi

8. Sequences and Series

easy

If the sum of the series $54 + 51 + 48 + .............$ is $513$, then the number of terms are

A

$18$

B

$20$

C

$17$

D

None of these

Solution

(a) According to condition

$ \Rightarrow $ $513 = \frac{n}{2}\left\{ {2 \times 54 + (n – 1)( – 3)} \right\}$

$ 1026=n(111-3n)$

$ \Rightarrow $ $3{n^2} – 111n + 1026 = 0$

$ (3n-57)(n-18)=0$

Hence $n = 18$.

Standard 11

Mathematics

Share

Similar Questions

If ${S_n} = nP + \frac{1}{2}n(n – 1)Q$, where ${S_n}$ denotes the sum of the first $n$ terms of an $A.P.$, then the common difference is

easy

If the sum of $n$ terms of an $A.P.$ is $\left(p n+q n^{2}\right),$ where $p$ and $q$ are constants, find the common difference.

medium

The sum of all those terms, of the anithmetic progression $3,8,13, \ldots \ldots .373$, which are not divisible by $3$,is equal to $…….$.

hard

(JEE MAIN-2023)

The interior angles of a polygon with n sides, are in an $A.P.$ with common difference $6^{\circ}$. If the largest interior angle of the polygon is $219^{\circ}$, then $n$ is equal to______

hard

(JEE MAIN-2025)

Let $a_1, a_2, a_3, \ldots$ be in an arithmetic progression of positive terms.

Let $\mathrm{A}_{\mathrm{k}}=\mathrm{a}_1{ }^2-\mathrm{a}_2{ }^2+\mathrm{a}_3{ }^2-\mathrm{a}_4{ }^2+\ldots+\mathrm{a}_{2 \mathrm{k}-1}{ }^2-\mathrm{a}_{2 \mathrm{k}}{ }^2$.

If $\mathrm{A}_3=-153, \mathrm{~A}_5=-435$ and $\mathrm{a}_1{ }^2+\mathrm{a}_2{ }^2+\mathrm{a}_3{ }^2=66$, then $\mathrm{a}_{17}-\mathrm{A}_7$ is equal to………………..

hard

(JEE MAIN-2024)