How to Sum a Linear Number Sequence (Summing a Sequence)

To work out the sum of some of the first few numbers in a linear sequence the following formula can be applied:

Sum = ½n[2a + d(n-1)]

Note:

a is the first number in the number sequence.

d is the common difference of the number sequence.

n is amount of terms that you need to total.

Example 1

Work out the sum of the first 30 numbers in this number sequence:

3, 8, 13, 18, 23…

First of all write down the values of a, d and n.

a = 3, as the first number in the number sequence is 3.

d = 5, as the sequence is increasing by 5 each time.

n = 30, as you are finding the sum of the first 30 numbers in the sequence.

Therefore, just plug these values into the formula above:

Sum = ½n[2a + d(n-1)]

Sum = ½ × 30[2×3 + 5(30-1)]

Sum = 15[6 + 5×29]

Sum = 15 × 151

Sum = 2265

Example 2

Work out the sum of the first 21 numbers in this number sequence:

11, 15, 19, 23, 27…

Again, write down the values of a, d and n.

a = 11, as the first number in the number sequence is 11.

d = 4, as the sequence is increasing by 4 each time.

n = 21, as you are finding the sum of the first 21 numbers in the sequence.

Therefore, just plug these values into the formula above:

Sum = ½n[2a + d(n-1)]

Sum = ½ × 21[2 × 11 + 5(21-1)]

Sum = 10.5[22 + 5×20]

Sum = 10.5 × 122

Sum = 1281

For some more examples on working out the sum of an increasing number sequence click here.

For some more examples on working out the sum of a decreasing number sequences click here.

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