Write the value of n for which nth of the A.P.s 3, 10, 17, ... and 63,

1.	Write the value of n for which nth of the A.P.s 3, 10, 17, ... and 63, 65, 67, ... are equal.
Answer» Let n^th term of A.P. 3,10,17,… be a_n And n^th term of A.P. 63,65,67,…be a_n' So,a₁ = 3 and a₂ = 10 ∴ d = a₂ - a₁ = 10 - 3 = 7 And, a₁'= 63 and a₂' = 65 ∴ d' = a₂' - a₁' = 65 - 63 = 2 Now, a_n = a₁+(n-1)d = 3+(n-1)7 = 3+7n-7 = 7n-4 And, a_n'=a₁'+(n-1)d' = 63+(n-1)² = 63+2n-2 = 2n+61 Now, It is given that a_n = a_n' ∴ 7n-4 = 2n+61 ∴ 7n-2n = 61+4 ∴ 5n = 65 ∴ n = 13

Write the value of n for which nth of the A.P.s 3, 10, 17, ... and 63, 65, 67, ... are equal.

Answer»

Let n^th term of A.P. 3,10,17,… be a_n

And n^th term of A.P. 63,65,67,…be a_n'

So,a₁ = 3 and a₂ = 10

∴ d = a₂ - a₁

= 10 - 3

= 7

And,

a₁'= 63 and a₂' = 65

∴ d' = a₂' - a₁' = 65 - 63 = 2

Now,

a_n = a₁+(n-1)d

= 3+(n-1)7

= 3+7n-7

= 7n-4

And,

a_n'=a₁'+(n-1)d'

= 63+(n-1)²

= 63+2n-2

= 2n+61

Now,

It is given that a_n = a_n'

∴ 7n-4 = 2n+61

∴ 7n-2n = 61+4

∴ 5n = 65

∴ n = 13