Let an arithmetic progression have 2n terms. The sum of the terms at odd positions is 230 and the sum of the terms at even positions is 250 . If the difference between the last term and the first term is 38 , then the first term of the arithmetic progression is equal to

Step-by-step solution

Let the arithmetic progression be a₁, a₂, a₃, , a_ 2n with common difference d . Given the sum of odd-positioned terms: S_ odd = a₁ + a₃ + + a_ 2n-1 = 230 And the sum of even-positioned terms: S_ even = a₂ + a₄ + + a_ 2n = 250 Subtracting S_ odd from S_ even : S_ even - S_ odd = (a₂ - a₁) + (a₄ - a₃) + + (a_ 2n - a_ 2n-1 ) 250 - 230 = d + d + + d ( n times) nd = 20 Also given the difference between the last and first terms: a_ 2n - a₁ = 38 (2n - 1)d = 38 2nd - d = 38 Substituting nd = 20 : 2(20) - d = 38 40 - d = 3