Question

Find the common difference of the A.P. and write the next two terms:

(i) $51,59,67,75,....$

Answer 1

Step 1. Note the given data:

Given A.P. is $51,59,67,75,....$

First term $t_1=51$

Second term $t_2=59$

Third term $t_3=67$

Fourth term $t_4=75$

Let $a$ be the first term of the A.P. and $d$ is the common difference of the A.P.

We need to find the common difference and the next two terms of A.P.

Step 2. Finding the common difference of given A.P:

Formula, the common difference of A.P.$={(n+1)}^{th}term-n^{th}term$

From the Arithmetic Progression, we get

Common difference($d$) $=secondtermofA.P.-firsttermofA.P.$

$$\begin{gathered} =59-51 \\ =8 \end{gathered}$$

Similarly,

$d=t_3-t_2=t_4-t_3=8$

Step 3. Finding the fifth term of A.P:

The $n^{th}$term of an AP is $t_n=a+(n-1)d$.

Here, $a=51,n=5$, and $d=8$.

The fifth term of A.P. $=a+4d$

$$\begin{gathered} =51+\left(4\times 8\right) \\ =51+32 \\ =83 \end{gathered}$$

Step 4. Finding the sixth term of A.P:

The $n^{th}$term of an AP is $t_n=a+(n-1)d$.

Here, $a=51,n=6$, and $d=8$.

The sixth term of A.P. $=a+5d$

$$\begin{gathered} =51+\left(5\times 8\right) \\ =51+40 \\ =91 \end{gathered}$$

So, the common difference of the given A.P. is $8$ and the next required terms of A.P. are $83,91$.