In an A.P., the 5^th and 9^th terms are 4 and -12 respectively. Find the:

(a) first term

(b) common difference

(c) sum of the first 20 terms.

If first term is a and common difference is d of the A.P.

By formula,

n^th term = a_n = a + (n - 1)d

Given,

⇒ 5^th term = 4

⇒ a₅ = 4

⇒ a + (5 - 1)d = 4

⇒ a + 4d = 4 ………(1)

⇒ 9^th term = -12

⇒ a₉ = -12

⇒ a + (9 - 1)d = -12

⇒ a + 8d = -12 ………(2)

Subtracting equation (1) from (2), we get :

⇒ (a + 8d) - (a + 4d) = -12 - 4

⇒ a - a + 8d - 4d = -16

⇒ 4d = -16

⇒ d = $-\dfrac{16}{4}$

⇒ d = -4.

Substituting value of d in equation (1), we get :

⇒ a + 4d = 4

⇒ a + 4(-4) = 4

⇒ a - 16 = 4

⇒ a = 4 + 16 = 20.

(a) Hence, first term of A.P. = 20.

(b) Common difference of A.P. = -4.

(c) By formula,

Sum of n terms of A.P. = $\dfrac{n}{2}(a + l)$

Sum of first 20 terms of A.P. = $\dfrac{n}{2}(a + a_{20})$

$= \dfrac{20}{2}[a + a + (20 - 1)d] \\[1em] = 10(2a + 19d) \\[1em] = 10 \times (2 \times 20 + 19 \times -4) \\[1em] = 10 \times (40 - 76) \\[1em] = 10 \times -36 \\[1em] = -360.$

Hence, sum of 20 terms of A.P. = -360.