The sum of the first n terms of an arithmetic sequence a1 a2

The sum of the first n terms of an arithmetic sequence a_1, a_2, a_3, ..., a_n is 1/2 n (a_1 + a_n), where a_1, and a_n are the first and the nth terms l n of the sequence, respectively. What is the sum of the odd integers from 1 to 99, inclusive?

Solution

If a is the first term of an arithmetic series and n is the common difference, the nth termof the series is a + ( n - 1)d.

In the given arithmetic series ( 1,3,5,7,9,11,....,99), the first term is 1 and the common difference is 2. Let us assume that 99 is the nth term of the seies. Then 99 = 1 + ( n - 1 )2 or, 99 = 2n - 1 or 2n = 100 or n = 50. Thus 99 is the 50th term of the given arithmetic series. The sum of this finite arithmetic seies is n/2( a₁ + a_n). Here, n = 50, a₁ = 1 and a_n = 99. Therefore, the sum of the odd integers from 1 to 99, inclusive is 50/2 ( 1 + 99) = 25*100 = 2500. The answer is D.