Question

Find the sum of all the integers between 1 to 50 which are not divisible by 3 ​

Answer 1

Step-by-step explanation:

The sum of first n numbers = \dfrac{n(n+1)}{2}

2

n(n+1)

So the sum of first 50 numbers = \dfrac{50(50+1)}{2}=1275

2

50(50+1)

=1275

The numbers are divisible by 3 = 3 , 6 , 9 , 12, ....,48

It follows Arithmetic progression with common difference = 3

nth term = a(n)= a+(n-1)d , where a= first term and d= common difference

Since last term is 48

So 48= 3+(n-1)(3)48=3+(n−1)(3)

48= 3+3n-348=3+3n−3

48=3n48=3n

n=16n=16 (Divide both sides by 16)

Sum of first n terms in AP= S_n=\dfrac{n}{2}(a+l)S

n

=

2

n

(a+l)

, where a= first term and l= last term

Put a= 3 , l =48 and n= 16

S_{16}=\dfrac{16}{2}(3+48)=8\times51=408S

16

=

2

16

(3+48)=8×51=408

i.e. Sum of all numbers from 1 to 50 that are divisible by 3 =408

The sum of all the integers between 1 to 50 which are not divisible by 3 =

sum of first 50 numbers - Sum of all numbers from 1 to 50 are divisible by 3

= 1275-408

=867

The sum of all the integers between 1 to 50 which are not divisible by 3 is 867 .