In an Arithmetic Progression (A.P.), the sum of the first 10 terms is 120 and the sum of t… — JAMB UTME

In an Arithmetic Progression (A.P.), the sum of the first $10$ terms is $120$ and the sum of the next $10$ terms is $320$ . What is the common difference of the progression?

A
$1$
$2$ Answer
C
$4$
D
$20$

Answer

The common difference of the progression is

2

The sum of an A.P. is

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

. For the first 10 terms,

S_{10} = 5(2a + 9d) = 120

, simplifying to

2a + 9d = 24

. The sum of the first 20 terms is

120 + 320 = 440

, so

S_{20} = 10(2a + 19d) = 440

, simplifying to

2a + 19d = 44

. Subtracting the two equations gives

10d = 20

, which yields the common difference

d = 2

Step-by-Step Solution

Express the sum of the first 10 terms using the formula

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

S_{10} = \frac{10}{2}[2a + 9d] = 5(2a + 9d) = 120 \implies 2a + 9d = 24

Relating the given sum of the first 10 terms to the first term

a

and common difference

d

generates the first linear equation.

Determine the sum of the first 20 terms (

S_{20}

) and set up the second linear equation.

S_{20} = S_{10} + \text{sum of next 10 terms} = 120 + 320 = 440

. Thus,

S_{20} = \frac{20}{2}[2a + 19d] = 10(2a + 19d) = 440 \implies 2a + 19d = 44

The sum of the next 10 terms added to the sum of the first 10 terms gives the total sum of the first 20 terms.

Solve the system of simultaneous linear equations for

d

(2a + 19d) - (2a + 9d) = 44 - 24 \implies 10d = 20 \implies d = 2

Subtracting equation (1) from equation (2) eliminates

a

to solve directly for the common difference

d

Key Concept

Sum of an Arithmetic Progression and Simultaneous Equations

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