Question

If the sum of first n terms of an AP is 5n^2+2n then the third term is

Answer 1

The sum of n terms of an AP with first term a and common difference d is
S = n{2a+(n-1)d}/2 = an + n²d/2 -nd/2
S = (d/2)n² + (a - d/2)n

The question​ is provided with the expression of sum to be,

S = 5n² +2 n

Both of these expressions must give the same value. So, the coefficients of n² and n, in both the expressions must be same.

Coefficients of n²:

d/2 = 5
d = 10

Coefficients of n:

2 = a - d/2 = a - 10/2 = a -5
2 = a-5
a = 7

Hence, a= 7 and d =10

3rd term = a +2d = 7+2×10 = 27

I hope you understand the approach.
All the best!

Answer 2

$\huge \underline{ \underline{ \purple{ \tt Solution :}}}$

$\sf given \: \: \: that \: \: : \bf{an} = {5n}^{2} + 2n$

then ,

$\bf{a3} = 5( {3)}^{2} + 2(3) = 5(9) + 6 = 45 +6 = 51$

$\bf{ \therefore} \: \: \sf{a3(third \: \: term) \: = 51}$

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