Name: Sequence and series
Uploaded: 2019-07-19
Duration: 23 min 56 s
Description: In this video, the basic concepts of Sequence and series and sum of n terms of series are discussed. After watching this video, you will be able to solve most of the questions related to this topic.

Question

If in an arithmetic progression $p^{th}$ term is $\frac{1}{q}$ and $q^{th}$ term is $\frac{1}{p}$ , then prove that $(p q)^{th}$ term of arithmetic progression is $1$ .

Accepted Answer

We know that, $p^{th}$ term of an AP is $\frac{1}{q}$

Where, $a =$ first term of an AP

$d =$ common difference

$a_{n} = n^{th}$ term f an AP.

Given: $p^{th}$ term $= \frac{1}{q}$ and $q^{th}$ term $= \frac{1}{p}$

$\Rightarrow a_{p} = \frac{1}{q} = a + (p - 1) d . . . . (i)$

$a_{q} = \frac{1}{p} = a + (q - 1) d . . . . (ii)$

Subtracting $(i)$ from $(ii)$ , we get

$\{a + (p - 1) d\} - \{a + (q - 1) d\} = \frac{1}{q} - \frac{1}{p}$

$\Rightarrow a + p d - d - a - q d + d = \frac{1}{q} - \frac{1}{p}$

$\Rightarrow p d - q d = \frac{1}{q} - \frac{1}{p}$

$\Rightarrow d (p - q) = \frac{p - q}{q p}$

$\Rightarrow d = \frac{p - q}{q p (p - q)}$

$\Rightarrow d = \frac{1}{p q}$

Now, substituting $d = \frac{1}{p q}$ in equation $(i)$ , we get

$\frac{1}{q} = a + (p - 1) \times \frac{1}{p q}$

$\Rightarrow \frac{1}{q} = a + \frac{p}{p q} - \frac{1}{p q}$

$\Rightarrow \frac{1}{q} = a + \frac{1}{q} - \frac{1}{p q}$

$\Rightarrow a = \frac{1}{p q}$

Therefore, $a_{p q} = \frac{1}{p q} + (p q - 1) \times \frac{1}{p q}$

$\Rightarrow a_{p q} = \frac{1}{p q} + 1 - \frac{1}{p q}$

$\Rightarrow a_{p q} = 1$

Therefore, ${(p q)}^{th}$ term of the given AP is $1$ .

If in an arithmetic progression $p^{th}$ term is $\frac{1}{q}$ and $q^{th}$ term is $\frac{1}{p}$ , then prove that $(p q)^{th}$ term of arithmetic progression is $1$ .

Important Questions on Arithmetic Progression

Learn and Prepare for any exam you want !

If in an arithmetic progression pth term is 1q and qth term is 1p, then prove that (pq)th term of arithmetic progression is 1.

Important Questions on Arithmetic Progression

Learn and Prepare for any exam you want !

If in an arithmetic progression $p^{th}$ term is $\frac{1}{q}$ and $q^{th}$ term is $\frac{1}{p}$ , then prove that $(p q)^{th}$ term of arithmetic progression is $1$ .