Question

The sum of the 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 ?

Answer 1

we are given that the sum of the first six terms is 42 . and using the sum to n terms formula ${S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]$ we get the equation (1) and we are that the 10 th term and 30th term are in the ratio 1:3.Using this we get a = dand substituting in equation (1) we get the value of a which is our first term

Complete step-by-step answer:
We are given that the sum of the first six terms of the AP is 42
Let the AP be a , a+d , a+2d, ……
With a as the first term and d as the common difference
We know that the formula to find the sum of n terms is given by
$\Rightarrow {S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]$
Here as we are given the sum of first six terms our n = 6
$\Rightarrow 42 = \dfrac{6}{2}\left[ {2a + (6 - 1)d} \right] \\\ \Rightarrow 42 = 3\left[ {2a + 5d} \right] \\\ \Rightarrow 2a + 5d = \dfrac{{42}}{3} \\\$
$\Rightarrow 2a + 5d = 14$ …………(1)
We are given that the ratio of its 10th term to its 30th term is 1 : 3
Therefore the nth term is given by
$\Rightarrow {a_n} = a + (n - 1)d$
Therefore the 10th term and 30th term is given by
$\Rightarrow {a_{10}} = a + 9d \\\ \Rightarrow {a_{30}} = a + 29d \\\$
As we are given the ratio is 1 : 3
$\Rightarrow \dfrac{{a + 9d}}{{a + 29d}} = \dfrac{1}{3}$
Cross multiplying we get
$\Rightarrow 3(a + 9d) = a + 29d \\\ \Rightarrow 3a + 27d = a + 29d \\\ \Rightarrow 3a + 27d - a - 29d = 0 \\\ \Rightarrow 2a - 2d = 0 \\\$
$\Rightarrow a = d$ ……..(2)
Substituting in (1)
$\Rightarrow 2a + 5d = 14 \\\ \Rightarrow 2a + 5a = 14 \\\ \Rightarrow 7a = 14 \\\ \Rightarrow a = \dfrac{{14}}{7} \\\ \Rightarrow a = 2 \\\$
Hence now we get that a = 2
That is the first term of the AP is 2.

Note: In an Arithmetic Sequence the difference between one term and the next is a constant. We can find the common difference of an AP by finding the difference between any two adjacent terms.If we know the initial term, the following terms are related to it by repeated addition of the common difference. Thus, the explicit formula is
Term = initial term + common difference * number of steps from the initial term