Question

Check if given series is $APs$? If they form an $AP$, find the common difference $d$ and write three more terms.
$2,\frac{5}{2},3,\frac{7}{2}....$

Answer 1

The given series is $2,\frac{5}{2},3,\frac{7}{2},\cdots$.

$a_2-a_1=\frac{5}{2}-2$

$=\frac{5-4}{2}$

$=\frac{1}{2}$

$a_3-a_2=3-\frac{5}{2}$

$=\frac{6-5}{2}$

$=\frac{1}{2}$

$a_4-a_3=\frac{7}{2}-3$

$=\frac{7-6}{2}$

$=\frac{1}{2}$

This shows that the difference is same, so the given series is in A.P. with common difference $\frac{1}{2}$. The next three terms are,

$a_5=\frac{7}{2}+\frac{1}{2}$

$=\frac{8}{2}$

$=4$

$a_6=4+\frac{1}{2}$

$=\frac{8+1}{2}$

$=\frac{9}{2}$

$a_7=\frac{9}{2}+\frac{1}{2}$

$=\frac{10}{2}$

$=5$