Arithmetic

Progressions

• Sum Based on relationship

between a_n and S_n

11) If the sum of the first n terms of an AP is 4n – n²,What is the first term ? What is the sum of the first two terms? What is the second term? Similarly, find the 3^rd, 10^th and n^th terms.

Sol:

S_n = 4n – n²

S₁ =

4

(1)

= 4 – 1

– (1)²

= 3

a₁ =

S₁ =

3

S₂ =

4

(2)

= 8 - 4

- (2)²

= 4

a₂ =

S₂ – S₁

= 4 - 3

= 1

d = a₂ – a₁

= 1 – 3

= - 2

a₃ =

a + 2d =

3 + 2(-2)

= 3 – 4

= – 1

a₁₀ =

a + 9d =

3 + 9(-2)

= 3 – 18

= – 15

a_n =

a + (n – 1) d

=

3

+ (n – 1)

(-2)

=

3

– 2n + 2

∴ a_n =

5 – 2n

Exercise 5.3 11