Question

if sn=5n(squared)+3n.then find 15th term?

Answer 1

here is your answer
hopes helps you dearr✌️✌️

Answer 2

Answer:

$\bold{148}$

Step-by-step explanation:

$First term \ = a \\ \\ Common difference \ = d \\ \\ No. of terms \ = n \\ \\ n^{th}\ term \ = \ T_n \\ \\ Sum of first \ n\ terms \ = \ S_n$

$\\ \\ \\ In any AP, the sum of first n digits is in the form of \ \ \frac{d}{2}n^2 + (a - \frac{d}{2})n. \\ \\ S_n = \frac{d}{2}n^2 + (a - \frac{d}{2})n$

$Here, \\ \\ \frac{d}{2}n^2 + (a - \frac{d}{2})n = 5n^2 + 3n \\ \\ \\ Coefficient of \ n^2\ in both sides :- \\ \\ \frac{d}{2} = 5 \\ \\ d = 5 \times 2 = 10 \\ \\ \\ Coefficient of \ n\ in both sides :- \\ \\ a - \frac{d}{2} = 3 \\ \\ = a - 5 = 3 \\ \\ a = 3 + 5 = 8$

$We got common difference and first term. \\ \\ \therefore\ The AP is \ \ 8, 18, 28, ......$

$\\ \\ 15^{th}\ term \\ \\ = T_{15} \\ \\ = a + 14d \\ \\ = 8 + 14 \times 10 \\ \\ = 8 + 140 = \bold{148} \\ \\ \\ \therefore\ 15^{th}\ term is \ \bold{148}$

$OR \\ \\ When sum of first \ n - 1\ terms is subtracted from sum of first \ n\ terms, then we get the \ n^{th}\ term. \\ \\ \therefore \\ \\ T_n =\ S_n -\ S_{n - 1}$

$\\ \\ = 5n^2 + 3n - [5(n - 1)^2 + 3(n - 1)] \\ \\ = 5n^2 + 3n - [5(n^2 - 2n + 1) + 3n - 3] \\ \\ = 5n^2 + 3n - [5n^2 - 10n + 5 + 3n - 3] \\ \\ = 5n^2 + 3n - 5n^2 + 10n - \bold{5} - 3n + 3 \\ \\ = 10n - 5 + 3 \\ \\ = 10n - 2$

$So we got the algebraic expression. \\ \\ 15^{th}\ term \\ \\ = T_{15} \\ \\ = 10 \times 15 - 2 \\ \\ = 150 - 2 = \bold{148} \\ \\ \\ \therefore\ 15^{th}\ term is \ \bold{148}$

$Let me prefer another simple method. \\ \\ S_n = 5n^2 + 3n \\ \\ S_1 = \ T_1 = 5(1)^2 + 3(1) = 5 + 3 = 8 \\ \\ S_2 =\ T_1 \ +\ T_2 \\ \\ = 5(2)^2 + 3(2) \\ \\ = 5 \times 4 + 3 \times 2 \\ \\ = 20 + 6 = 26 \\ \\ S_2\ - \ S_1 =\ T_1\ + \ T_2\ - \ T_1 = \ T_2 \\ \\ = 26 - 8 = 18$

$\\ \\ d = \ T_2\ - \ T_1 = 18 - 8 = 10 \\ \\ So we have first term and common difference. Then let's find the 15^{th}\ term. \\ \\ T_{15} \\ \\ = a + 14d \\ \\ = 8 + 14 \times 10 \ \ \ \ \ \ \ \ \ \ [\ a = \ T_1 \ ] \\ \\ = 8 + 140 = \bold{148} \\ \\ \\ \therefore\ 15^{th}\ term is \ \bold{148}$

$\\ \\ \\ This method is so simple and suitable to you if you're weak in algebra.$

$Hope this may be helpful.\\ \\ Please mark my answer as the \ \bold{brainliest}\ if this may be helpful. \\ \\ Thank you. Have a nice day. \\ \\ \\ \#adithyasajeevan$