Question

Show that a sequence is an ap if its 10th term is a linear expression in n and in such a case the common difference is equal to the coefficient of n

Answer 1

Correct question

Show that a sequence is an A.P. if its nth term is a linear expression in n and in such a case the common difference is equal to the coefficient of n

Proof

Let nth term of a sequence be expressed with linear expression a + bn

This means that nth term = a + bn

⇒ (n + 1)th term = a + b(n + 1)

⇒ (n + 1)th term = a + bn + b

and, (n - 1)th term = a + b(n - 1)

⇒ (n - 1)th term = a + bn - b

Now, we know that three terms x, y and z are in A.P. if

2y = z + x

hence, if (n - 1)th term, nth term, (n + 1)th term are i A.P., then

2(nth term) = (n + 1)th term + (n - 1)th term

We will check it for a + bn - b, a + bn and a + bn + b

2(a + bn) = 2a + 2bn

and,

a + bn - b + a + bn + b = 2a + 2bn

⇒ 2(a + bn) = a + bn - b + a + bn + b

hence, the three terms (a + bn - b), (a + bn) and (a + bn + b) are in A.P.

⇒ Sequence with nth term expressed as a + bn is an A.P.

⇒ A sequence with its nth term expressed as a linear expression in n is an A.P.

Now, common difference = (n + 1th)term - nth term

here, (n + 1)th term = a + bn + b and nth term = a + bn

⇒ common difference = a + bn + b - (a + bn)

⇒ a + bn + b - a - bn

= b

Here, b is the coefficient of n

hence, the common difference of an A.P. expressed as a linear expression of n is the coefficient of n.

Proved.