The sum of first m terms of an AP is (4m2 - m). If its nth term is 107, find the value of n. Also, find the 21st term of this AP.

Let a be the first term and d be the common difference.

Given: Sum of first m terms of an AP is given by:

Sm = [2a + (m - 1)d] = 4m2 - m

Now, nth term is given by: an = Sn - Sn - 1

an = (4n2 - n) - [4(n - 1)2 - (n - 1)]

= (4n2 - n) - [4(n2 + 1 - 2n) - n + 1]

= 4n2 - n - 4n2 - 4 + 8n + n - 1

= 8n - 5 …………………(1)

Now, given that an = 107

8n - 5 = 107

8n = 112

n = 14

For 21st term of AP, put n = 21 in the value of the nth term in equation (1), we get

a21 = 8 × (21) - 5

a21 = 168 - 5

= 163

a21 = 163

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