Question

Find the common difference and write the next four terms of each of the following arithmetic progression:
$-1,-\frac{5}{6},-\frac{2}{3},...$

Answer 1

Given A.P is

$-1,-\frac{5}{6},-\frac{2}{3},....$

the common difference of an A.P is given by $d=a_{n+1}-a_n$

by putting n=1 in above equation

$d=a_2-a_1=-\left(\frac{5}{6}\right)-(-1)$

$d=-\frac{5}{6}+1$

$d=\frac{1}{6}$

first term of this A.P is

$a_1=-1$

the nth term of this A.P is given by

$a_n=a_1+(n-1)d$

$⟹a_n=-1+(n-1)\left(\frac{1}{6}\right)$

$⟹a_n=-1+\frac{1}{6}n-\frac{1}{6}$

$⟹a_n=-\frac{7}{6}+\frac{1}{6}n\dots .eq\left(1\right)$

finding next four terms of A.P

1) for finding the 4th term of sequence $\left(a_4\right)$ put n=4 in eq(1)

$⟹a_4=-\frac{7}{6}+\frac{1}{6}\times 4$

$⟹a_4=-\frac{7}{6}+\frac{4}{6}$

$⟹a_4=-\frac{3}{6}$

$⟹a_4=-\frac{1}{2}$

2) for finding the 5th term of sequence $\left(a_5\right)$ put n=5 in eq(1)

$⟹a_5=-\frac{7}{6}+\frac{1}{6}\times 5$

$⟹a_5=-\frac{7}{6}+\frac{5}{6}$

$⟹a_5=-\frac{2}{6}$

$⟹a_5=-\frac{1}{3}$

3) for finding the 6th term of sequence $\left(a_6\right)$ put n=6 in eq(1)

$⟹a_6=-\frac{7}{6}+\frac{1}{6}\times 6$

$⟹a_6=-\frac{7}{6}+\frac{6}{6}$

$⟹a_6=-\frac{1}{6}$

4) for finding the 7th term of sequence $\left(a_7\right)$ put n=7 in eq(1)

$⟹a_7=-\frac{7}{6}+\frac{1}{6}\times 7$

$⟹a_7=-\frac{7}{6}+\frac{7}{6}$

$⟹a_7=0$