# Ex.5.2 Q14 Arithmetic Progressions Solution - NCERT Maths Class 10

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## Question

How many multiples of $$4$$ lie between $$10$$ and $$250$$?

Video Solution
Arithmetic Progressions
Ex 5.2 | Question 14

## Text Solution

What is Known?

Numbers between $$10$$ and $$250$$

What is Unknown?

Multiples of $$4$$ between $$10$$ and $$250.$$

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:

By Observation, First multiple of $$4$$ that is greater than $$10$$ is $$12.$$

Next will be $$16.$$

Therefore, the series will be as follows: $$12,\, 16,\, 20,\, 24,\, \dots$$

All these are divisible by $$4$$ and thus, all these are terms of an A.P. with first term as $$12$$ and common difference as $$4.$$

When we divide $$250$$ by $$4,$$ the remainder will be $$2.$$ Therefore, $$250 – 2 = 248$$ is divisible by $$4$$ which is the largest multiple of $$4$$ within $$250.$$

Hence the final series is as follows:

$12,\, 16,\, 20,\, 24,\, \dots, 248$

Let $$248$$ be the $$n\rm{th}$$ term of this A.P.

We know that the $$n\rm{th}$$ term of an A.P. Series,

\begin{align}a &= 12\\d &= 4\\{a_n} &= 248\\{a_n} &= a + (n - 1)d\\248 &= 12 + (n - 1)4\\\frac{{236}}{4} &= n - 1\\n& = 60\end{align}

Therefore, there are $$60$$ multiples of $$4$$ between $$10$$ and $$250.$$

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