To find the sum of an arithmetic sequence, first we add all the terms and take it as the first equation. Then, we
reverse the terms and add both the equations. On solving, we get the equation of the sum of the given arithmetic
sequence.

Let a = the first term of the series, d = the common difference, an = the last term and s = the sum of all the given terms.

Then s = a + (a + d) + (a + 2d) + (a + 3d) +………..+ a + (n-2) d + a + (n-1) d

By reversing the above equation, we get

s = a + (n-1) d + a + (n-2) d +…………. + (a + 3d) + (a + 2d) + (a + d) + a

Adding the above two equations, we get

2s = [2a + (n-1) d] + [2a + (n-1) d] + [2a + (n-1) d] +……..up to n terms.

2s = n [2a + (n-1) d]

s = n/2 [2a + (n-1) d]

s = n/2 [a + a + (n-1) d]

s = n/2 (a + an)

Hence the sum of the sequence is:

s = n/2 {2a + (n-1) d} or s = n/2 {a + an}

Example

Find the sum of first ten terms of the sequence 3, 8, 13, 18………

Solution    a = 3

                    d = 8-3 = 5

                    n = 10

By using the above formula of the sum of a sequence

s = n/2 {2a + (n-1) d}

s = 10/2 {2*3 + (10-1)5}

s = 5 {6 + 45}

s = 5*51

s = 255

Try these questions

1. What is the sum of the first 50 positive integers from the sequence in question 9?

   (a) 1125

   (b) 1565

   (c) 1345

   (d) 1275

   Answer: D

   a = 1 an = 50 n = 50

   s = n/2 {a + an}

   s = 50/2 (1 + 50) = 1275




2. What is the sum of the first five terms of the arithmetic sequence 9, 17, 25……?

   (a) 100

   (c) 150

   (d) 125

   Answer: D

   a = 9, d = 17-9 = 8

   s = n/2 {2a + (n -1) d}

   s = 5/2 {2*9 + (5 -1) 8}

   s = 125




3. Find Σ (2n + 1) where n =1, 2, 3…….6. (Note: Σ means sum.)

   (a) 48

   (b) 56

   (c) 84

   (d) 60

   Answer: A

   s = n/2 {a + an}

   n = 6

   a = (2*1 + 1) = 3

   an = (2*6 + 1) = 13

   Hence, s = 48