Question

Two A. p shave the same common difference. The first term of one A. p is 2 and that of the other is 7, The difference between their ${10}^{t}h$ term is the same as the difference between their 21^st term, which is the same as the difference any two corresponding terms. why?

Answer 1

Let two A.P.s have first terms & difference be a, d & a', d' respectively.

As $a_n=a+(n-1)d$; $a=2$, $a^{\prime }=7$

& $a_{10}-a_{10}^{\prime }=a_{21}-a_{21}^{\prime }$

$\Rightarrow (a+9d)-(a^{\prime }+9d^{\prime })=(a+20d)-(a^{\prime }+20d^{\prime })$

$\Rightarrow 9(d-d^{\prime })=20(d-d^{\prime })$

$\Rightarrow 11(d-d^{\prime })=0$

$\Rightarrow d=d^{\prime }$.