How many terms are there in the A.P.?

In the given problem, we are given an A.P.

We need to find the number of terms present in it

So here we will find the value of $n$ using the formula, $a_{n}=a+(n-1) d$

(i) Here, A.P is $7,10,13, \ldots .43$

The first term $(a)=7$

The last term $\left(a_{n}\right)=43$

Now,

Common difference $(d)=a_{1}-a$

$=10-7$

$=3$

Thus, using the above mentioned formula, we get,

$43=7+(n-1) 3$

$43-7=3 n-3$

$36+3=3 n$

$n=\frac{39}{3}$

$n=13$

Thus, $n=13$

Therefore, the number of terms present in the given A.P is 13

(ii) Here, A.P is $-1,-\frac{5}{6},-\frac{2}{3},-\frac{1}{2}, \ldots, \frac{10}{3}$

The first term $(a)=-1$

The last term $\left(a_{n}\right)=\frac{10}{3}$

Now,

Common difference $(d)=a_{1}-a$

$=-\frac{5}{6}-(-1)$

$=-\frac{5}{6}+1$

$=\frac{-5+6}{6}$

$=\frac{1}{6}$

Thus, using the above mentioned formula, we get,

$\frac{10}{3}=-1+(n-1) \frac{1}{6}$

$\frac{10}{3}+1=\frac{1}{6} n-\frac{1}{6}$

$\frac{13}{3}+\frac{1}{6}=\frac{1}{6} n$

Further solving for n, we get

$\frac{26+1}{6}=\frac{1}{6} n$

$n=\frac{27}{6}(6)$

$n=27$

Thus, $n=27$

Therefore, the number of terms present in the given A.P is 27

(iii) Here, A.P is $7,13,19, \ldots .205$

The first term $(a)=7$

The last term $\left(a_{n}\right)=205$

Now,

Common difference $(d)=a_{1}-a$

$=13-7$

$=6$

Thus, using the above mentioned formula, we get,

$205=7+(n-1) 6$

$205-7=6 n-6$

$198+6=6 n$

$n=\frac{204}{6}$

$n=34$

Thus, $n=34$

Therefore, the number of terms present in the given A.P is 34

(iv) Here, A.P is $18,15 \frac{1}{2}, 13, \ldots,-47$

The first term $(a)=18$

The last term $\left(a_{n}\right)=-47$

Now,

Common difference $(d)=a_{1}-a$

$=15 \frac{1}{2}-18$

$=\frac{31}{2}-18$

$=\frac{31-36}{2}$

$=-\frac{5}{2}$

Thus, using the above mentioned formula, we get,

$-47=18+(n-1)\left(-\frac{5}{2}\right)$

$-47-18=-\frac{5}{2} n+\frac{5}{2}$

$-65-\frac{5}{2}=-\frac{5}{2} n$

Further, solving for n, we get

$\frac{-130-5}{2}=-\frac{5}{2} n$

$-\frac{135}{2}(2)=-5 n$

$n=\frac{-135}{-5}$

$n=27$

Thus, $n=27$

Therefore, the number of terms present in the given A.P is 27 .