Question

the first terms of the two parallel series are equal and the ratio of common differences is 1:2 If the 7th term of first A.P is 23 and 21th term of second A.P is 125 then,find the first term?​

Answer 1

Answer:

First term = 5 .

Step-by-step explanation:

We are given that the first terms of the two parallel series are equal and the ratio of common differences is 1 : 2.

Let the first term of both AP series be a and the common difference of first AP series be $d_1$ and that of second AP series be $d_2$ .

Also, it is given that 7th term of first A.P is 23 and 21th term of second A.P is 125 which means;  $a_7$ = 23    and   $a_2_1$ = 125

 ⇒  a + (7 - 1)*$d_1$ = 23                 and           a + (21 - 1)*$d_2$ = 125

 ⇒  a + 6*$d_1$ = 23                       and            a + 20*$d_2$ = 125

 ⇒  a = 23 - 6*$d_1$ ---[Equation 1]      and            a = 125 - 20*$d_2$ -----[Equation 2]

Equating both equations we get,

         ⇒  23 - 6*$d_1$ = 125 - 20*$d_2$

         ⇒  20*$d_2$ - 6*$d_1$ = 102

         ⇒  $20*\frac{d_2}{d_2} - 6*\frac{d_1}{d_2} = \frac{102}{d_2}$  {by dividing whole equation by $d_2$}

         ⇒  20 - 6 * $\frac{1}{2}$ = $\frac{102}{d_2}$   {because ratio of common differences is 1:2}

         ⇒   $d_2 = \frac{102}{17}$ = 6

So, putting this value of $d_2$ in equation 2 we get ;

                a = 125 - 20 * 6 = 5

Hence, first term = 5.