# Ex.5.3 Q14 Arithmetic Progressions Solution - NCERT Maths Class 10

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## Question

Find the sum of the odd numbers between $$0$$ and $$50.$$

Video Solution
Arithmetic Progressions
Ex 5.3 | Question 14

## Text Solution

What is Known?

Odd numbers between $$0$$ and $$50$$

What is Unknown?

Sum of the odd numbers between $$0$$ and $$50$$

Reasoning:

Sum of the first

$$n$$ terms of an AP is given by $${S_n} = \frac{n}{2}\left[ {a + l} \right]$$, and $$n\rm{th}$$ term of an AP is $$\,{a_n} = a + \left( {n - 1} \right)d$$ Where $$a$$ is the first term, $$d$$ is the common difference and $$n$$ is the number of terms and $$l$$ is the last term.

Steps:

The odd numbers between $$0$$ and $$50$$ are $$1, \,3,\, 5,\, 7,\, 9 \,... \,49$$

Therefore, it can be observed that these odd numbers are in an A.P.

Hence,

• First term, $$a = 1$$
• Common difference, $$d = 2$$
• Last term, $$l = 1$$

We know that nth term of AP, $$\,{a_n} = l = a + \left( {n - 1} \right)d$$

\begin{align}49 &= 1 + \left( {n - 1} \right)2\\48& = 2\left( {n - 1} \right)\\n - 1 &= 24\\n &= 25 \end{align}

We know that sum of $$n$$ terms of AP,

\begin{align}{S_n} &= \frac{n}{2}\left( {a + l} \right)\\{S_{25}} &= \frac{{25}}{2}\left( {1 + 49} \right)\\ &= \frac{{25}}{2} \times 50\\& = 25 \times 25\\ &= 625\end{align}

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