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.
Given: nth term of the A.P. 9, 7, 5,… is same as the nth term of the A.P. 15, 12, 9 …
We know, an = 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, an = a1 + (n – 1)d1
⇒ an = 9 + (n – 1)(-2)
⇒ an = 9 – 2n + 2
⇒ an = 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, bn = b1 + (n – 1)d2
⇒ bn = 15 + (n – 1)(-3)
⇒ bn = 15 – 3n + 3
⇒ bn = 12 – 3n
According to question:
an = bn
⇒ 11 – 2n = 12 – 3n
⇒ 3n – 2n = 12 – 11
⇒ n = 1
Hence, the value of n is 1