Question

The first term of two A.P.s are equal and the ratios of their common
differences is 1 : 2. If the 7th term
of first A.P. and 21th term
of
second A.P. are 23 and 125 respectively. Find two A.P.S.​

Answer 1

Answer:

First AP = 5, 8, 11, 14, 17,........

Second AP = 5, 11, 17, 23,........

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 }

         ⇒  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  in equation 2 we get ;

                a = 125 - 20 * 6 = 5  and $d_1$ = 6/2 = 3

Hence, First AP series = a, a+$d_1$, a+2*$d_1$,.......

                                     = 5, 5+3, 5+2*3,.......

                                     = 5, 8, 11, 14, 17,........

Second AP series = a, a+$d_2$, a+2*$d_2$,...........

                              = 5, 11, 17, 23,............... {because $d_2$ = 6}