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Question: If the sum of \(n\) terms of an A.P. is \(2 n ^ { 2 } + 5 n\) , then the \(n ^ { t h }\) term will b...

If the sum of nn terms of an A.P. is 2n2+5n2 n ^ { 2 } + 5 n , then the nthn ^ { t h } term will be.

A

4n+34 n + 3

B

4n+54 n + 5

C

4n+64 n + 6

D

4n+74 n + 7

Answer

4n+34 n + 3

Explanation

Solution

Given that Sn=2n2+5nS _ { n } = 2 n ^ { 2 } + 5 n

Putting n=1,2,3,………,S1=2×1+5×1=7n = 1,2,3 , \ldots \ldots \ldots , S _ { 1 } = 2 \times 1 + 5 \times 1 = 7 ,

S2=2×4+10=8+10=18,S3=18+15=33S _ { 2 } = 2 \times 4 + 10 = 8 + 10 = 18 , S _ { 3 } = 18 + 15 = 33.

So, T1=S1=a=7,T2=S2−S1=18−7=11T _ { 1 } = S _ { 1 } = a = 7 , T _ { 2 } = S _ { 2 } - S _ { 1 } = 18 - 7 = 11,

T3=S3−S2=33−18=15T _ { 3 } = S _ { 3 } - S _ { 2 } = 33 - 18 = 15

Therefore series is

Now, nthn ^ { t h } term

=a+(n−1)d=7+(n−1)4=4n+3= a + ( n - 1 ) d = 7 + ( n - 1 ) 4 = 4 n + 3 .

Aliter : As we know

Tn=Sn−Sn−1T _ { n } = S _ { n } - S _ { n - 1 }

=(2n2+5n)−{2(n−1)2+5(n−1)}= \left( 2 n ^ { 2 } + 5 n \right) - \left\{ 2 ( n - 1 ) ^ { 2 } + 5 ( n - 1 ) \right\}

=2n2+5n−2n2+4n−2−5n+5=4n+3= 2 n ^ { 2 } + 5 n - 2 n ^ { 2 } + 4 n - 2 - 5 n + 5 = 4 n + 3