The 3^{\text{rd}} term of an arithmetic progression (AP) is 10 and the 7^{\text{th}} term … — JAMB UTME

The $3^{\text{rd}}$ term of an arithmetic progression (AP) is $10$ and the $7^{\text{th}}$ term is $22$ . What is the sum of the first $12$ terms of the progression?

A
$228$
$246$ Answer
C
$264$
D
$444$

Answer

The sum of the first

12

terms of the arithmetic progression is

246

Using the

n^{\text{th}}

term formula

T_n = a + (n-1)d

, the equations

a + 2d = 10

and

a + 6d = 22

yield

d = 3

and

a = 4

. Substituting these values into

S_{12} = \frac{12}{2}[2(4) + 11(3)]

gives

6 \times 41 = 246

Step-by-Step Solution

Set up simultaneous equations using the

n^{\text{th}}

term formula

T_n = a + (n-1)d

a + 2d = 10

and

a + 6d = 22

The

3^{\text{rd}}

term corresponds to

n=3

and the

7^{\text{th}}

term corresponds to

n=7

Subtract the first equation from the second to find the common difference

d

4d = 12 \implies d = 3

Subtracting eliminates the first term

a

Substitute

d = 3

back into the first equation to find the first term

a

a + 2(3) = 10 \implies a = 4

Determining the first term is necessary to calculate the sum.

Calculate the sum of the first

12

terms using

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

S_{12} = \frac{12}{2}[2(4) + (12-1)(3)] = 6[8 + 33] = 6(41) = 246

Applying the AP sum formula with

n = 12

a = 4

, and

d = 3

Key Concept

Arithmetic Progression: Finding common difference, first term, and sum of terms

Estimated Time:1m 30s

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