Question
Write the $n^{th}$ ter, of the A.P. $\frac{1}{\text{m}},\frac{1+\text{m}}{\text{m}},\frac{1+2\text{m}}{\text{m}}, .....$
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Answer
Given: A.P. $\frac{1}{\text{m}},\frac{1+\text{m}}{\text{m}},\frac{1+2\text{m}}{\text{m}}, .....$
We know that the $n^{th}$ term of an A.P. is given by
$a_n=a+(n-1) d$
In the given A.P.
$\text{a}=\frac{1}{\text{m}}$
$\text{d}=\frac{1+\text{m}}{\text{m}}-\frac{1}{\text{m}}=\frac{1+\text{m}-1}{\text{m}}=1$
Thus, the $n^{th}$ term of the given A.P. is
$\text{a}_\text{n}=\frac{1}{\text{m}}+(\text{n}-1)1=\frac{1+(\text{n}-1)\text{m}}{\text{m}}$
We know that the $n^{th}$ term of an A.P. is given by
$a_n=a+(n-1) d$
In the given A.P.
$\text{a}=\frac{1}{\text{m}}$
$\text{d}=\frac{1+\text{m}}{\text{m}}-\frac{1}{\text{m}}=\frac{1+\text{m}-1}{\text{m}}=1$
Thus, the $n^{th}$ term of the given A.P. is
$\text{a}_\text{n}=\frac{1}{\text{m}}+(\text{n}-1)1=\frac{1+(\text{n}-1)\text{m}}{\text{m}}$
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