How to calculate the common difference when the first term and the nth term is know.

Question:

The 28th term of an AP is -5. Find the common difference if its first term is 31.

Solution

T1 = a ( First term of the sequence)

a = 31

T28 = -5 (28th term of the sequence)

d = ? (common difference)

Recall the nth term formula of Ap

i.e Tn = a + (n-1)d

Substitute 28 for n to get the following equation

T28 = a + (28-1)d

T28 = a + 27d

Substitute T28 for -5 and 31 for a in the equation above

-5 = 31 + 27d

Subtract 31 from both sides of the equation

-5 -31 = 27d

-36 = 27d

Divide through by 27

-36/27 = 27d/27

d = -36/27

d = -4/3

The common difference is -4/3

CHECK

Substitute the first term and the common difference into the equation below,

T28 = a + 27d

T28 = 31 + 27 * -4/3

Divide 27 by 3 to get 9

T28 = 31 - 9 * 4

T28 = 31 - 36

T28 = - 5

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