The sum of the first four terms of an arithmetic progression (A.P.) is 32, and the sum of … — JAMB UTME

The sum of the first four terms of an arithmetic progression (A.P.) is $32$ , and the sum of the next four terms is $96$ . What is the common difference of the progression?

Answer: 4

Answer

The common difference of the arithmetic progression is

4

The sum of the first four terms yields

2a + 3d = 16

. The sum of the first eight terms is

32 + 96 = 128

, which gives

2a + 7d = 32

. Subtracting these two linear equations gives

4d = 16

, leading to

d = 4

Step-by-Step Solution

Formulate an equation for the sum of the first 4 terms.

2a + 3d = 16

The sum of the first

n

terms of an A.P. is

S_n = \frac{n}{2}(2a + (n-1)d)

. Substituting

n = 4

and

S_4 = 32

gives

2(2a + 3d) = 32

, which simplifies to

2a + 3d = 16

Formulate an equation for the sum of the first 8 terms.

2a + 7d = 32

The total sum of the first 8 terms is the sum of the first 4 terms plus the sum of the next 4 terms (

S_8 = 32 + 96 = 128

). Substituting

n = 8

gives

4(2a + 7d) = 128

, which simplifies to

2a + 7d = 32

Solve the system of simultaneous linear equations for

d

d = 4

Subtracting

(2a + 3d = 16)

from

(2a + 7d = 32)

eliminates

2a

, resulting in

4d = 16

, which yields

d = 4

Key Concept

Sum of an Arithmetic Progression

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