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# Ex.5.2 Q15 Arithmetic Progressions Solution - NCERT Maths Class 10

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## Question

For what value of n, are the nth terms of two APs $$63,\, 65,\, 67,\,$$and $$3,\, 10,\, 17,\,\dots$$ equal?

Video Solution
Arithmetic Progressions
Ex 5.2 | Question 15

## Text Solution

What is Known:?

Two different APs

What is Unknown?

Value of $$n$$ so that two APs have equal $$n^\rm{th}$$ 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:

If $$n\rm{th}$$ terms of the two APs $$63, \,65,\, 67,\dots$$ and $$3,\, 10,\, 17,\, \dots$$ are equal.

Then,

\begin{align} 63 + \left( {n-1} \right)2 =& 3 + \left( {n-1} \right)7 \\& \, \dots \text{Equation (1)}\end{align}

[Since In $$1\rm{st}$$ AP, $$a = 63$$, $$d = 65--3 = 2$$ and in $$2\rm{nd}$$ AP , $$a = 3$$,$$d = 10--3 = 7$$]

By Simplifying Equation (1)

\begin{align}7\left( {n-1} \right)-2\left( {n-1} \right)&= 63-3\\7n - 7 - 2n + 2 &= 60\\5n - 5 &= 60\\n &= \frac{{65}}{5}\\n &= 13\end{align}

The $$13^\rm{th}$$ terms of the two given APs are equal.

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