Question

The fifth term of an AP is 20 and the sum of its seventh and eleventh term is 64. Find the common difference​

Answer 1

Given :-

• fifth term of AP , a₅ = 20

• sum of seventh and eleventh term of AP is 64 , a₇ + a₁₁ = 64

To find :-

• common difference of AP

Solution :-

As given

↦ a₅ = 20

↦ a + 4 d = 20

↦ a = 20 - 4 d ..... equation (1)

Also,

↦ a₇ + a₁₁ = 64

↦ a + 6 d + a + 10 d = 64

↦ 2 a + 16 d = 64

using equation (1)

↦ 2 ( 20 - 4 d ) + 16 d = 64

↦ 40 - 8 d + 16 d = 64

↦ 8 d + 40 = 64

↦  8 d = 64 - 40

↦ 8 d = 24

↦ d = 24 / 8

↦ d = 3

Hence,

Common difference of AP is 3 .

____________________________

• A sequence of numbers having difference between their consecutive terms constant , is known as A.P. (Arithmetic progression).

• nth term of an AP is represented as ,      aₙ = a + ( n - 1 ) d  

• Sum of first n terms of an AP is given by , Sₙ = n ( 2 a + ( n - 1 ) d ) / 2

( where a is first term , and d is the common difference of A.P. )

____________________________

Answer 2

GIVEN:

• The fifth term of an AP is 20 and the sum of its seventh and eleventh term is 64.

TO FIND:

• Common difference = ?

SOLUTION:

→$\sf a_5 = 20$

→$\sf a + 4d = 20....(1)$

→$\sf a_7 + a_{11} = 64$

→$\sf a + 6d + a + 10d = 64$

→$\sf 2a + 16d = 64$

→$\sf 2 \left(a + 8d\right) = 64$

→$\sf a + 8d = \dfrac{64}{2}$

→$\sf a + 8d = 32.....(2)$

By solving both equations we get,

→$\underline{\boxed{\gray{ \textbf{d = 3}}}}$ $\red\bigstar$

Therefore, Common Difference is 3.