The sum of first six terms of an A.P. is 42. The ratio of its 10th term to its 30th term is 1:3. Find the first term of the A.P.

Question

Philoid · Accepted Answer

If 'a' is the first term and 'd' is the common difference in an AP,

We know that,

nth term of an AP, a_n = a + (n - 1)d

Sum of first 'n' terms of an AP,

Now, given

Sum of first six terms, S₆ = 42

⇒ 3[2a + 5d] = 42

⇒ 2a + 5d = 14 …[1]

Also, given ratio of 10^th and 30^th term is 1 : 3

⇒ 3a + 27d = a + 29d

⇒ 2a = 2d

⇒ a = d

Putting d = a in [1]

⇒ 2a + 5a = 14

⇒ 7a = 14

⇒ a = 2

i.e. First term of AP is 2.

The sum of first six terms of an A.P. is 42. The ratio of its 10^th term to its 30^th term is 1:3. Find the first term of the A.P.