If the nth term of the A.P. 9, 7, 5,… is same as the nth term of the

1.	If the nth term of the A.P. 9, 7, 5,… is same as the nth term of the A.P. 15, 12, 9 … Find n.
Answer» Given : n^th term of the A.P. 9, 7, 5,… is same as the nth term of the A.P. 15, 12, 9 … We know, a_n = a + (n – 1)d Where a is first term or a1 and d is common difference and n is any natural number Let A.P. 9, 7, 5,… has first term a₁ and common difference d₁ ⇒ a₁ = 9 and a₂ = 7 Common difference, d₁ = a₂ – a₁ = 7 – 9 = -2 Now, a_n = a₁ + (n – 1)d₁ ⇒ a_n = 9 + (n – 1)(-2) ⇒ a_n = 9 – 2n + 2 ⇒ a_n = 11 – 2n Let A.P. 15, 12, 9 … has first term a₁ and common difference d₁ ⇒ b₁ = 15 and b₂ = 12 Common difference, d₂ = b₂ – b₁ = 12 – 15 = -3 Now, b_n = b₁ + (n – 1)d₂ ⇒ b_n = 15 + (n – 1)(-3) ⇒ b_n = 15 – 3n + 3 ⇒ b_n = 12 – 3n According to question : a_n = b_n ⇒ 11 – 2n = 12 – 3n ⇒ 3n – 2n = 12 – 11 ⇒ n = 1 Hence, The value of n is 1.

If the nth term of the A.P. 9, 7, 5,… is same as the nth term of the A.P. 15, 12, 9 … Find n.

Answer»

Given :

n^th term of the A.P. 9, 7, 5,… is same as the nth term of the A.P. 15, 12, 9 …

We know,

a_n = a + (n – 1)d

Where a is first term or a1 and d is common difference and n is any natural number

Let A.P. 9, 7, 5,… has first term a₁ and common difference d₁

⇒ a₁ = 9 and a₂ = 7

Common difference,

d₁ = a₂ – a₁

= 7 – 9

= -2

Now,

a_n = a₁ + (n – 1)d₁

⇒ a_n = 9 + (n – 1)(-2)

⇒ a_n = 9 – 2n + 2

⇒ a_n = 11 – 2n

Let A.P. 15, 12, 9 … has first term a₁ and common difference d₁

⇒ b₁ = 15 and b₂ = 12

Common difference,

d₂ = b₂ – b₁

= 12 – 15

= -3

Now,

b_n = b₁ + (n – 1)d₂

⇒ b_n = 15 + (n – 1)(-3)

⇒ b_n = 15 – 3n + 3

⇒ b_n = 12 – 3n

According to question :

a_n = b_n

⇒ 11 – 2n = 12 – 3n

⇒ 3n – 2n = 12 – 11

⇒ n = 1

Hence,

The value of n is 1.