# Ex.5.1 Q2 Arithmetic progressions Solutions - NCERT Maths Class 10

## Question

Write first four terms of AP, When the first term \(a\) and the common difference \(d\) are given as follows:

(i) \(\begin{align} a = 10,\,d = 10\end{align}\)

(ii) \(\begin{align}a= - 2\quad , d = 0\end{align}\)

(iii) \(\begin{align}a= 4,\, d= - 3\end{align}\)

(iv) \(\begin{align}a = - 1,\,d = \frac{1}{2}\end{align}\)

(v) \(\begin{align}a = - 1.25,\,d = - 0.25\end{align}\)

## 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)** \(\begin{align} a = 10,\,d = 10\end{align}\)

**What is Known?**

\(\begin{align} a= 10\,\, {\rm{and}}\,\, d = 10\end{align}\)

**What is Unknown?**

First four terms of the AP.

**Steps:**

First term \(\begin{align}{a = 10}\end{align}\)

Second term \(\begin{align}{a+d = 10 + 10 = 20}\end{align}\)

Third term \(\begin{align}{a + 2d = 10 + 20 = 30}\end{align}\)

Fourth term \(\begin{align}{a + 3d = 10 + 30 = 40}\end{align}\)

The first four terms of AP are \(10, 20, 30,\) and \(40.\)

**(ii)** \(\begin{align}a= - 2,\quad d = 0\end{align}\)

**What is Known?**

\(\begin{align}a= - 2\,\, {\rm{and}}\,\, d = 0\end{align}\)

**What is Unknown?**

First four terms of the AP.

**Steps:**

First term \(\begin{align}{a =} -2\end{align}\)

Second term \(\begin{align}= a+ {d} = -2+0 = -2 \end{align}\)

Third term \(\begin{align}= a + {2d} = -2 +0 = -2 \end{align}\)

Fourth term \(\begin{align}= a + {3d} = -2 + 0 = -2 \end{align}\)

The first four terms of AP are \(-2,-2,-2\) and \(-2.\)

** (iii)** \(\begin{align}a= 4,\, d= - 3\end{align}\)

**What is Known?**

\(\begin{align} {a = 4}\,\, {\rm{and}}\,\, {d =} -3 \end{align}\)

**What is Unknown?**

First four terms of the AP.

**Steps:**

First term \(\begin{align} {= a = 4} \end{align}\)

Second term \(\begin{align} { a+d = 4+ (-3) = 1} \end{align}\)

Third term \(\begin{align} { a+2d} = 4 – 6= -2\end{align}\)

Fourth term \(\begin{align} { a+3d} = 4\; – 9 = -5\end{align}\)

The first four terms of AP are \(\begin{align} 4, 1, -2, -5\end{align}\).

**(iv) **\(\begin{align}a = - 1,\,d = \frac{1}{2}\end{align}\)

**What is Known?**

\(\begin{align}a = -1\,\, {\rm{and}}\,\, {d} =\frac{1}{2}\end{align}\)

**What is Unknown?**

First four terms of the AP.

**Steps:**

First term = \(a = - 1\)

Second term\(\begin{align}= {{a}} + {{d}} = - 1 + \frac{1}{2} = - \frac{1}{2}\end{align}\)

Third term \(\begin{align}= {{a}} + 2{{d}} = - 1 + 1 = 0\end{align}\)

Fourth term \(\begin{align}= {{a}} + 3{{d}} = - 1 + \frac{3}{2} = \frac{1}{2}\end{align}\)

The first four terms of AP are \(\begin{align} - 1, - \frac{1}{2},0,\frac{1}{2}\end{align}\) .

**(v)** \(\begin{align}a = - 1.25,\,d = - 0.25\end{align}\)

**What is Known?**

\(\begin{align}{a}= -1.25\,\, {\rm{and}}\,\, \quad {d} = - 0.25\end{align}\)

**What is Unknown?**

First four terms of the AP.

**Steps:**

First term\(\begin{align} = a = - 1.25\end{align}\)

Second term

\(\begin{align}&= {{a}} + {{d}}\\ &= - 1.25 + ( - 0.25)\\ &= - 1.25 - 0.25\\ &= - 1.5\\ \end{align}\)

Third term

\(\begin{align}&= a + 2d\\ &= - 1.25 + 2 \times ( - 0.25)\\ &= - 1.25 - 0.50\\ &= - 1.75\end{align}\)

Fourth term

\(\begin{align}&= a + 3{{d}}\\ &= - 1.25 + 3 \times ( - 0.25)\\ &= - 1.25 - 0.75\\ &= - 2.00\end{align}\)

The first four terms of AP are

\(\begin{align}-1.25, -1.5, -1.75, \rm{and} - 2.00\end{align}\).