Question

How many multiples of 3 lie between 20 and 205?
A. 59
B. 61
C. 62
D. 63

Answer 1

Hint: In this type of question we can find starting multiple of 3 and last multiple of 3 in the given range of numbers. From multiples we can find Arithmetic series because there is a common difference between two consecutive terms. Then we can use general term of formula of A.P to get number of terms in A.P

Complete step-by-step answer:
Multiple of 3 just after 20 is 21.
Multiple of 3 just less than 205 is 204.
So we can write multiple of 3 in form of series as below:
 $ 21,24,27,30,...............................,201,204 $
First term is (a) = 21
Common difference (d) is = 24-21 = 27-24 = 30-27 = 3
Last term is 204.
Let the number of terms in series is n.
So we can write $ {{T}_{n}}=204 $
In general nth term of A.P is
 $ {{T}_{n}}=a+(n-1)d $
So we can write it as
 $ a+(n-1)d=204 $
On substituting value of a and d
 $ \Rightarrow 21+(n-1)\times 3=204 $
 $ \Rightarrow 21+3n-3=204 $
 $ \Rightarrow 3n+18=204 $
 $ \Rightarrow 3n=204-18 $
 $ \Rightarrow 3n=186 $
 $ \Rightarrow n=\dfrac{186}{3} $
 $ \Rightarrow n=62 $
Hence there are 62 multiples of 3 between 20 and 205.
Therefore option C is the correct answer.

Note:Multiples of 3 are that number which we can divide by 3.
If first term of an A.P is a , common difference is d and number of term is n then general term of A.P is
 $ {{T}_{n}}=a+(n-1)d $