Question
Find the common difference and write the next four terms of the following arithmetic progressions:
$-1,\frac{5}{6},\frac{2}{3},\ .....$
$-1,\frac{5}{6},\frac{2}{3},\ .....$
✓
Answer
Here, first term$ (a_1) = -1$
Common difference $(d) = a_2 - a_1$
$=-\frac{5}{6}-(-1)$
$=\frac{-5+6}{6}$
$=\frac{1}{6}$
Now, we need to find the next four terms of the given A.P
That is we need to find $a_4, a_5, a_6, a_7.$
So, using the formula $a_n = a + (n - 1)d$
Substituting n = 4, 5, 6, 7 in the above formula
Substituting n = 4, we get
$\text{a}_4=-1+(4-1)\Big(\frac{1}{6}\Big)$
$\text{a}_4=-1+\Big(\frac{1}{2}\Big)$
$\text{a}_4=\frac{-2+1}{2}=\frac{-1}{2}$
Substituting n = 5, we get
$\text{a}_5=-1+(5-1)\Big(\frac{1}{6}\Big)$
$\text{a}_5=-1+\frac{2}{3}$
$\text{a}_5=\frac{-3+2}{3}$
$\text{a}_5=-\frac{1}{3}$
Substituting n = 6, we get
$\text{a}_6=-1+(6-1)\Big(\frac{1}{6}\Big)$
$\text{a}_6=-1+\frac{5}{6}$
$\text{a}_6=\frac{-6+5}{6}$
$\text{a}_6=-\frac{1}{6}$
Substituting n = 7, we get
$\text{a}_7=-1+(7-1)\Big(\frac{1}{6}\Big)$
$\text{a}_7=-1+1$
$\text{a}_7=0$
Therefore, the common difference is $\text{d}=\frac{1}{6}$ and the next four terms are $-\frac{1}{2},-\frac{1}{3},-\frac{1}{6},0$.
Common difference $(d) = a_2 - a_1$
$=-\frac{5}{6}-(-1)$
$=\frac{-5+6}{6}$
$=\frac{1}{6}$
Now, we need to find the next four terms of the given A.P
That is we need to find $a_4, a_5, a_6, a_7.$
So, using the formula $a_n = a + (n - 1)d$
Substituting n = 4, 5, 6, 7 in the above formula
Substituting n = 4, we get
$\text{a}_4=-1+(4-1)\Big(\frac{1}{6}\Big)$
$\text{a}_4=-1+\Big(\frac{1}{2}\Big)$
$\text{a}_4=\frac{-2+1}{2}=\frac{-1}{2}$
Substituting n = 5, we get
$\text{a}_5=-1+(5-1)\Big(\frac{1}{6}\Big)$
$\text{a}_5=-1+\frac{2}{3}$
$\text{a}_5=\frac{-3+2}{3}$
$\text{a}_5=-\frac{1}{3}$
Substituting n = 6, we get
$\text{a}_6=-1+(6-1)\Big(\frac{1}{6}\Big)$
$\text{a}_6=-1+\frac{5}{6}$
$\text{a}_6=\frac{-6+5}{6}$
$\text{a}_6=-\frac{1}{6}$
Substituting n = 7, we get
$\text{a}_7=-1+(7-1)\Big(\frac{1}{6}\Big)$
$\text{a}_7=-1+1$
$\text{a}_7=0$
Therefore, the common difference is $\text{d}=\frac{1}{6}$ and the next four terms are $-\frac{1}{2},-\frac{1}{3},-\frac{1}{6},0$.
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