# Ex.5.2 Q12 Arithmetic Progressions Solution - NCERT Maths Class 10

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## Question

Two APs have the same common difference. The difference between their $$100^\rm{th}$$ term is $$100,$$ what is the difference between their $$1000^\rm{th}$$ terms?

Video Solution
Arithmetic Progressions
Ex 5.2 | Question 12

## Text Solution

What is Known:?

Two APs with the same common difference and difference between their $$100^\rm{th}$$ term.

What is Unknown?

Difference between their $$1000^\rm{th}$$ term

Reasoning:

$${a_n} = a + \left( {n - 1} \right)d$$ is the general term of AP. Where $${a_n}$$ is the $$n\rm{th}$$ term, $$a$$ is the first term, $$d$$ is the common difference and $$n$$ is the number of terms.

Steps:

Let the first term of these A.P.s be $${a_1}$$ and $${b_1}$$ respectively and the Common difference of these A.P’s be $$d$$

For first A.P.,

\begin{align}{a_{100}} &= {a_1} + (100 - 1)d\\ &= {a_1} + 99{\rm{d}}\\{a_{1000}} &= {a_1} + (1000 - 1)d\\{a_{1000}} &= {a_1} + 999d\end{align}

For second A.P.,

\begin{align}{b_{100}} &= {b_1} + (100 - 1)d\\& = {b_1} + 99d\\{b_{1000}}& = {b_1} + (1000 - 1)d\\ &= {{\rm{b}}_1} + 999d\end{align}

Given that, difference between

$$100^\rm{th}$$ term of these A.P.s $$= 100$$

Thus, we have

\begin{align}&\left( {{a_1} + 99d} \right) - \left( {{b_1} + 99d} \right)= 100\\&{a_1} - {b_1}= 100\quad \dots {\rm{Equation}}\left( 1 \right)\end{align}

Difference between $$1000^\rm{th}$$ terms of these A.P.s

\begin{align}&\left( {{a_1} + 999d} \right)\! -\! \left( {{b_1}\! +\! 999d} \right) \\&=\! {a_1}\! - \!{b_1}\quad\dots{\rm{Equation}}\left( 2 \right)\end{align}

From equation (1) & Equation (2),

This difference, $${a_1} - {b_1} = 100$$

Hence, the difference between $$1000^\rm{th}$$ terms of these A.P. will be $$100.$$

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