Question

sum of first 2n terms of an ap is alpha and the sum of next n terms is beta its common difference is​

Answer 1

Answer:

d = α - 2β / 3 n^2

Step-by-step explanation:

Given  

sum of first 2 n terms of an AP is alpha and the sum of next n terms is beta its common difference is

 We know that in an arithmetic progression first term is a and common difference is d. So in an A.P we have terms as a, a+d, a+3 d………….a + (n – 1)d.

Now sum to n terms is given by

Sn = n/2 [2 a + (n – 1)d]

So α = 2n/2(2a + (n – 1)d)

 α/n = 2 a + (2 n – 1)d

 2 a = α/n – (2n – 1) d-----------(1)

 Now α + β = 3 n/2 [2 a + (3 n – 1)d]

 2 a = 2(α + β)/3 n – (3 n – 1)d-----------(2)

 α / n – (2 n – 1)d = 2(α + β) / 3n – (3 n – 1)d

 α / n – 2(α + β)/3 n = (2 n – 1)d – (3 n – 1)d

1/n(α – 2(α + β)/3) = 2 nd – d – 3 nd + d

 1/n(α - 2 β/3) = - nd

1/3n(α - 2β) = - nd

α -2β = - 3n^2d

d = α - 2β / 3 n^2

Answer 2

Answer:

Step-by-step explanation: