Sum of first six terms of an AP is 42 and ratio of 10th term to its 30th term is 1:3. Calculate fist

Sum of first six terms of an AP is 42 and ratio of 10th term to its 30th term is 1:3. Calculate fist




Solution -

Tn = [a + (n-1)d]
Sn = n/2*[2a +(n-1)d].
Given
T30/T10 = 3
(a + 29d)/(a + 9d) = 3
After simplification we get
a = d

S6 = 6/2 [2a + 5d] = 42 (given)
14 = 2a + 5d
14/7 = 2 = a = d ( substituting a = d)

So the series starts like
2, 4, 6, 8, 10, 12, 14, 16, .......
Now T13 = [2+12d] = [2 + 24] = 26.