# Ex.5.2 Q9 Arithmetic Progressions Solution - NCERT Maths Class 10

## Question

If the \(3^\rm{rd}\) and the \(9^\rm{th}\) terms of an AP are \(4\) and \(-8,\) respectively, which term of this AP is zero.

## Text Solution

**What is Known:?**

\(3^\rm{rd}\) and \(9^\rm{th}\) term of AP,

**What is Unknown?**

Which term of AP is zero**. **

**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:**

Third term of the AP \(= 4\)

\[a + 2d = 4 \qquad \dots\left( 1 \right)\]

\(9^\rm{th}\) term of AP \(= -8\)

\[a + 8d = - 8 \qquad \ldots .\left( 2 \right)\]

Solving (1) and (2) for \(a\) and \(d\)

\[\begin{align}& \frac{\begin{align}& a\text{ }+\text{ }2d\text{ }=4 \\ & a\text{ }+\text{ }8d\text{ }=-8 \\

\end{align}}{-6d\text{ }=\text{ }12} \\ &\qquad \; d\text{ } =-2 \\\end{align}\]

Putting \(d = - 2\) in equation (1)

\[\begin{align}a - 4 = 4\\a = 8\end{align}\]

\[\begin{align}a + (n - 1)d &= 0\\8 + (n - 1)( - 2)& = 0\\n - 1 &= 4\\n &= 5\end{align}\]

\(5^\rm{th}\) term will be \(0.\)