Question

How many terms are there in the A.P., 7, 13, 19, …205?
A. 32
B. 33
C. 34
D. 35

Answer 1

Hint- Here, we will proceed by using the formula to calculate nth term of A.P. which is given by ${a_n} = {a_1} + (n - 1) \times d$ where ${a_1}$ is first term of the series, d is common difference,${a_n}$is nth term of A.P. and n is no. of terms in A.P.

Complete step-by-step answer:

Given sequence is 7, 13, 19, …205
Here, first term ${a_1}$= 7
Second term ${a_2}$= 13
Third term ${a_3}$= 19
nth term ${a_n}$= 205
Common difference $d = {a_2} - {a_1}$= 13-7 = 6
Also $d = {a_3} - {a_2}$= 19-13 =6
${a_n} = $205
Now by using the formula of nth term of A.P ${a_n} = {a_1} + (n - 1) \times d$…(i)
Substituting the values of ${a_1},{a_n},d$ in equation (i)
$\Rightarrow$ 205 = 7 + (n-1)$ \times $6
$\Rightarrow$ 205 = 7 +6n-6
$\Rightarrow$ 205 = 1+ 6n
$\Rightarrow$ 205-1= 6n
$\Rightarrow$ $\dfrac{{204}}{6}$= n
$\Rightarrow$ n = 34
So, Correct option is (C).

Note – In these type of problems , when first term ${a_1}$, second term ${a_2}$,third term ${a_3}$ and nth term ${a_n}$ is given then we calculate the common difference d as difference between successive consecutive terms and then we used this formula ${a_n} = {a_1} + (n - 1) \times d$ to calculate the number of terms in the given series of A.P.