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

In this problem, we are given different A.P. and we need to find the common difference of the A.P., along with the next two terms.

(i) $51,59,67,75, \ldots$

Here,

$a_{1}=51$

$a_{2}=59$

So, common difference of the A.P. $(d)=a_{2}-a_{1}$

$=59-51$

$=8$

Also, we need to find the next two terms of A.P., which means we have to find the $5^{\text {th }}$ and $6^{\text {th }}$ term.

So, for fifth term,

$a_{5}=a_{1}+4 d$

$=51+4(8)$

$=51+32$

$=83$

Similarly, we find the sixth term,

$a_{6}=a_{1}+5 d$

$=51+5(8)$

$=51+40$

$=91$

Therefore, the common difference is $d=8$ and the next two terms of the A.P. are $a_{5}=83, a_{6}=91$.

(ii) $75,67,59,51 \ldots$

Here,

$a_{1}=75$

$a_{2}=67$

So, common difference of the A.P. $(d)=a_{2}-a_{1}$

$=67-75$

$=-8$

Also, we need to find the next two terms of A.P., which means we have to find the $5^{\text {th }}$ and $6^{\text {th }}$ term.

So, for fifth term,

$a_{5}=a_{1}+4 d$

$=75+4(-8)$

$=75-32$

$=43$

Similarly, we find the sixth term,

$a_{6}=a_{1}+5 d$

$=75+5(-8)$

$=75-40$

$=35$

Therefore, the common difference is $d=-8$ and the next two terms of the A.P. are $a_{5}=43, a_{6}=35$.

(iii) $1.8,2.0,2.2,2.4, \ldots$

Here,

$a_{1}=1.8$

$a_{2}=2.0$

So, common difference of the A.P. $(d)=a_{2}-a_{1}$

$=2.0-1.8$

$=0.2$

Also, we need to find the next two terms of A.P., which means we have to find the $5^{\text {th }}$ and $6^{\text {th }}$ term.

So, for fifth term,

$a_{5}=a_{1}+4 d$

$=1.8+4(0.2)$

$=1.8+0.8$

$=2.6$

Similarly, we find the sixth term,

$a_{6}=a_{1}+5 d$

$=1.8+5(0.2)$

$=1.8+1$

$=2.8$

Therefore, the common difference is $d=0.2$ and the next two terms of the A.P. are $a_{5}=2.6, a_{6}=2.8$.

(iv) $0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \ldots$

Here,

$a_{1}=0$

$a_{2}=\frac{1}{4}$

So, common difference of the A.P. $(d)=a_{2}-a_{1}$

$=\frac{1}{4}-0$

$=\frac{1}{4}$

Also, we need to find the next two terms of A.P., which means we have to find the $5^{\text {th }}$ and $6^{\text {th }}$ term.

So, for fifth term,

$a_{5}=a_{1}+4 d$

$=0+4\left(\frac{1}{4}\right)$

$=1$

Similarly, we find the sixth term,

$a_{6}=a_{1}+5 d$

$=0+5\left(\frac{1}{4}\right)$

$=\frac{5}{4}$

Therefore, the common difference is $d=\frac{1}{4}$ and the next two terms of the A.P. are $a_{5}=1, a_{6}=\frac{5}{4}$.

(v) $119,136,153,170, \ldots$

Here,

$a_{1}=119$

$a_{2}=136$

So, common difference of the A.P. $(d)=a_{2}-a_{1}$

$=136-119$

$=17$

Also, we need to find the next two terms of A.P., which means we have to find the $5^{\text {th }}$ and $6^{\text {th }}$ term.

So, for fifth term,

$a_{5}=a_{1}+4 d$

$=119+4(17)$

$=119+68$

$=187$

Similarly, we find the sixth term,

$a_{6}=a_{1}+5 d$

$=119+5(17)$

$=119+85$

$=204$

Therefore, the common difference is $d=17$ and the next two terms of the A.P. are $a_{5}=187, a_{6}=204$.