Question

The first, tenth and twenty- eighth terms of an A.P. are three successive terms of a G.P.. Find the common ratio of the G.P.. Given that the sum of the first 28 terms of the A.P. is 210, find the first term.​

Answer 1

Arithmetic Progression & Geometric Progression

1st, 10th and 28th term of an AP are consecutive terms GP

By nth term of AP formula an = a + ( n - 1 )d

First term a1 = a

Tenth term a10 = a + 9d

Twenty eighth term a28 = a + 27d

In General,

If a', b, c are in GP then there exist a relation

b² = a'c

Here, according to the given condition b = a10, a' = a, c = a27

So, substitute them in the relation

( a + 9d )² = a( a + 27d )

a² + 81d² + 18ad = a² + 27ad

81d² = 27ad - 18ad

81d² = 9ad

9d = a

Sum of 28 terms of the AP = 210

Using Sum of n terms of an AP formula Sn = n/2 ( 2a + ( n - 1 ) d)

S28 = 28/2 [ 2a + 27d ] = 210

14[ 2a + 3( 9d ) ] = 210

14( 2a + 3a ) = 210

14( 5a ) = 210

a = 210 /( 14 × 5 )

a = 3

Find the value of a10

a10 = a + 9d = a + a = 2a = 2( 3 ) = 6

Common ratio of GP = b/a' = a10/a = 6/3 = 2

Therefore the common ratio of GP is 2 and first term is 3.