Ex.5.1 Q3 Arithmetic progressions Solutions - NCERT Maths Class 10

For the following APs, write the first term and the common difference:

i) \(\begin{align}3,1, - 1, - 3 \ldots \ldots \end{align}\)

ii) \(\begin{align}- 5, - 1,3,7 \ldots \ldots  \end{align}\)

iii) \(\begin{align}\frac{1}{3},\frac{5}{3},\frac{9}{3},\frac{{13}}{3} \ldots \ldots.\quad \quad \end{align}\)

iv) \(\begin{align}0.6,1.7,2.8,3.9 \ldots \end{align}\)

Text Solution

 

Reasoning:

General form of an arithmetic progression is \(\begin{align}{a, (a+d), (a+2d), (a+3d)…}\end{align}\) where \(a\) is the first term and \(d\) is the difference.

i) What is known?

\(3,1,-1,-3\) are in AP.

What is Unknown?

First term and the common difference of the AP.

Steps:

The AP is \(\begin{align} 3, 1 ,-1, -3.\end{align}\)

First term \(\begin{align}{a = 3}.\end{align}\)

Common difference

\[\begin{align} & = {a_2} - {a_1}\\&= 1 – 3\\&= -2\end{align}\]

First term is \(3\) and the common difference is \(-2.\)

ii) What is known?

\(\begin{align}-5, -1, 3, 7\end{align}\) are in AP

What is Unknown?

First term and the common difference of the AP.

Step:

The AP is \(\begin{align}-5, -1, 3, 7.\end{align}\)

First term\(\begin{align} a = - 5\end{align}\)

Common difference

\[\begin{align} & = {a_2} - {a_1}\\&= -1 – (-5)\\&= -1 + 5\\& = 4\end{align}\]

First term \(-5\) and the common difference is \(4.\)

iii) What is known?

\(\begin{align}\frac{1}{3},\frac{5}{3},\frac{9}{3},\frac{{13}}{3}\end{align}\) are in AP.

What is Unknown?

First term and the common difference of the AP.

Steps:

The AP is  \(\begin{align}\frac{1}{3},\frac{5}{3},\frac{9}{3},\frac{{13}}{3}\end{align}\)

First term \(\begin{align} a = \frac{1}{3}\end{align}\)

Common difference

\[\begin{align} &= {a_2} - {a_1}\\&= \frac{5}{3} - \frac{1}{3}\\&= \frac{{5 - 1}}{3}\\&= \frac{4}{3}\end{align}\]

First term is \(\begin{align}\frac{1}{3}\end{align}\) and the common difference is \(\begin{align}\frac{4}{3}\end{align}\)

iv) What is known?

\(\begin{align}{\rm{0}}{\rm{.6,}}\,\,{\rm{1}}{\rm{.7,}}\,\,{\rm{2}}{\rm{.8,}}\,\,{\rm{3}} {\rm{.9}}\end{align}\) are in AP.

What is Unknown?

First term and the common difference of the AP.

Steps:

The AP is  \(\begin{align} {\rm{0}}{\rm{.6,}}\,\,{\rm{1}}{\rm{.7,}}\,\,{\rm{2}}{\rm{.8,}}\,\,{\rm{3}}{\rm{.9}}\end{align}\)

First term \(\begin{align} a = 0.6 \end{align}\)

Common difference

\[\begin{align} &= {a_2} - {a_1}\\&= 1.7 – 0.6\\&= 1.1\end{align}\]

First term \(= 0.6\) and the common difference \(= 1.1\)