Question

For what value of n, the nth terms of the arithmetic progressions 63, 65, 67, … and 3, 10, 17, … are equal?

Answer 1

Let the term of the given progressions be t_n and t_n, respectively.

The first AP is 63, 65, 67,... 

Let its first term be a and common difference be d.

Then a = 63 and d = (65 - 63) = 2

So, its nth term is given by

t_n = a + (n - 1)d

\(\Rightarrow\) 63 + (n - 1) x 2

\(\Rightarrow\) 61 + 2n

The second AP is 3, 10, 17,... 

Let its first term be A and common difference be D. 

Then A = 3 and D = (10 - 3) = 7 

So, its nth term is given by

T_n = A + (n - 1)d

\(\Rightarrow\) 63 + (n - 1) x 2

\(\Rightarrow\) 61 + 2n

The second AP is 3, 10, 17,... 

Let its first term be A and common difference be D. 

Then A = 3 and D = (10 - 3) = 7 

So, its nth term is given by

T_n = A + (n - 1)D

\(\Rightarrow\) 3 + (n - 1) x 7

\(\Rightarrow\) 7n - 4

Now, t_n = T_n

\(\Rightarrow\) 61 + 2n = 7n - 4

\(\Rightarrow\) 65 = 5n

\(\Rightarrow\) n = 13

Hence, the l3 terms of the Al’s are the same.

Answer 2

Hope this will help you