Question

7. In an arithmetic series, the sum of the first 9 terms is 162 and the sum of the first 12 terms is 288. Determine the series.

Answer 1

2, 6, 10, 14, 18, .....

Explanation:

the sum to n terms of an arithmetic series is

`S_(n)` =
`(n)/(2)` [ 2a₁ + (n - 1)d ]

where a₁ is the first term and d the common difference

given S₉ = 162 , then

`(9)/(2)` (2a₁ + 8d) = 162

4.5(2a₁ + 8d) = 162 ( divide both sides by 4.5 )

2a₁ + 8d = 36 → (1)

given S₁₂ = 288 , then

`(12)/(2)` (2a₁ + 11d) = 288

6(2a₁ + 11d) = 288 ( divide both sides by 6 )

2a₁ + 11d = 48 → (2)

solve the 2 equations simultaneously to find a₁ and d

2a₁ + 8d = 36 → (1)

2a₁ + 11d = 48 → (2)

subtract (1) from (2) term by term to eliminate a₁

(2a₁ - 2a₁) + (11d - 8d) = 48 - 36

0 + 3d = 12

3d = 12 ( divide both sides by 3 )

d = 4

substitute d = 4 into either of the 2 equations and solve for a₁

substituting into (1)

2a₁ + 8(4) = 36

2a₁ + 32 = 36 ( subtract 32 from both sides )

2a₁ = 4 ( divide both sides by 2 )

a₁ = 2

then series is

a, a + d, a + 2d, .....

= 2, 6, 10, 14, 18, .......