Given:
a = 2, d = 8 and s_n = 90
Find the value of n and a₂.

Given: a = 2, d = 8 and Sn = 90

To find: Here we have to find the value of n and a₂.

Solution:

S_n = 90

$\frac{n}{2}[2a\ +\ (n\ -\ 1)d]\ =\ 90$

$\frac{n}{2} [2\ \times\ 2\ +\ (n\ -\ 1)8]\ = 90$

$n{4\ +\ 8n\ -\ 8}\ =\ 180$

$n{8n\ -\ 4}\ =\ 180$

$8n^2\ -\ 4n\ =\ 180$

$8n^2\ -\ 4n\ -\ 180\ =\ 0$

$2n^2\ -\ n\ -\ 45\ =\ 0$

$2n^2\ -\ 10n\ +\ 9n\ -\ 45\ =\ 0$

$2n(n\ -\ 5)\ +\ 9(n\ -\ 5)\ =\ 0$

$(2n\ +\ 9)(n\ -\ 5)\ =\ 0$

So,

n = $-\frac{9}{2}$ or n = 5

But n cannot be negative. Then,

n = 5

Now, calculating the value of a₂:

$a_{n}\ =\ a\ +\ (n\ -\ 1)d$

$a_{2}\ =\ 2\ +\ (2\ -\ 1)8$

$a_{2}\ =\ 2\ +\ (1)8$

$a_{2}\ =\ 2\ +\ 8$

$\mathbf{a_{n} \ =\ 10}$

So, value of n is 5 and a₂ is 10.

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Updated on: 10-Oct-2022

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