An arithmetic progression has an odd number of terms. The sum of all terms in odd positions is 266 , and the sum of all terms in even positions is 228 . If the last term of the arithmetic progression exceeds the first term by 48 , then the first term of the arithmetic progression is :

Step-by-step solution

Let the arithmetic progression have 2m+1 terms, with first term a and common difference d . The terms in odd positions are T₁, T₃, T₅, , T_ 2m+1 . There are m+1 such terms. The terms in even positions are T₂, T₄, T₆, , T_ 2m . There are m such terms. The middle term of the entire A.P. is T_ m+1 . For both the odd-positioned sub-sequence and the even-positioned sub-sequence, the average of the terms is equal to the middle term T_ m+1 . Therefore, the sum of the odd-positioned terms is: S_ odd = (m+1) T_ m+1 = 266 Th