Find the sum of the first:51 terms of the A.P. : whose second term is

1.	Find the sum of the first:51 terms of the A.P. : whose second term is 2 and fourth term is 8.
Answer» We know that the sum of terms for different arithmetic progressions is given by S_n = \(\frac{n}{2}\)[2a + (n − 1)d] Where; a = first term for the given A.P. d = common difference of the given A.P. n = number of terms 51 terms of an AP whose a₂ = 2 and a₄ = 8 We know that, a₂ = a + d 2 = a + d …(2) Also, a₄ = a + 3d 8 = a + 3d … (2) Subtracting (1) from (2), we have 2d = 6 d = 3 Substituting d = 3 in (1), we get 2 = a + 3 ⟹ a = -1 Given that the number of terms (n) = 51 First term (a) = -1 So, S_n = \(\frac{51}{2}\)[2(−1) + (51 − 1)(3)] = \(\frac{51}{2}\)[−2 + 150] = \(\frac{51}{2}\)[158] = 3774 Hence, the sum of first 51 terms for the A.P. is 3774.

Find the sum of the first:51 terms of the A.P. : whose second term is 2 and fourth term is 8.

Answer»

We know that the sum of terms for different arithmetic progressions is given by

S_n = \(\frac{n}{2}\)[2a + (n − 1)d]

Where; a = first term for the given A.P. d = common difference of the given A.P. n = number of terms

51 terms of an AP whose a₂ = 2 and a₄ = 8

We know that, a₂ = a + d

2 = a + d …(2)

Also, a₄ = a + 3d

8 = a + 3d … (2)

Subtracting (1) from (2), we have

2d = 6

d = 3

Substituting d = 3 in (1), we get

2 = a + 3

⟹ a = -1

Given that the number of terms (n) = 51

First term (a) = -1

So,

S_n = \(\frac{51}{2}\)[2(−1) + (51 − 1)(3)]

= \(\frac{51}{2}\)[−2 + 150]

= \(\frac{51}{2}\)[158]

= 3774

Hence, the sum of first 51 terms for the A.P. is 3774.