Question

The sum of n term of an A.P is given by Sn=2n2-3n then answer of the following questions
What is sum of the first n-terms
a. -1
b. 1
c. 2
d. 3​

Answer 1

EXPLANATION.

Sum of Nth term of Ap → Sn - S(n-1).

→ Sn = 2n² - 3n.

→ 2n² - 3n - [ 2( n - 1)² - 3( n - 1) ].

→ 2n² - 3n - [ 2 ( n² + 1 - 2n ) - 3n + 3 ].

→ 2n² - 3n - [ 2n² + 2 - 4n - 3n + 3 ].

→ 2n² - 3n - 2n² - 2 + 4n + 3n - 3 ].

→ 4n - 5 = Algebraic expression.

→ n = 1 → 4(1) - 5 = -1.

→ n = 2 → 4(2) - 5 = 3.

→ n = 3 → 4(3) - 5 = 7.

→ n = 4 → 4(4) - 5 = 11.

Series are = -1 , 3, 7, 11,........

First term = a = -1.

Common difference = d = b - a = 3 + 1 = 4.

Formula of Sum of Nth term.

$\sf \: s_{n} \: = \dfrac{n}{2} (2a \: + (n - 1)d)$

$\sf \: s_{n} \: = \dfrac{n}{2} ( - 2 + (n - 1)4) \\ \\ \sf \: s_{n} \: = \frac{n}{2} ( - 2 + 4n - 4) \\ \\ \sf \: s_{n} \: = \frac{n}{2}(4n - 6) \\ \\ \sf \: s_{n} \: = \frac{n}{2} \times 2(2n - 3)$

$\sf \: s_{n} \: = n(2n - 3) \\ \\ \sf \: s_{n} \: = 2 {n}^{2} - 3n \: = \: proved$