Step-by-step explanation:

Given :-

The numbers from 101 and 300

To find :-

The sum of all the non multiples of 5 from 101 to 300.

Solution :-

Given numbers = 101,102,103,...,300

First term (a) = 101

Common Difference (d) = 102-101 = 1

Number of terms (n) = 200

We know that

Sum of the first 'n' terms in an AP

=> (n/2)[2a+(n-1)d]

On substituting these values in the above formula then

=> (200/2)[2(101)+(200-1)(1)]

=> 100(202+199)

=> 100(401)

=> 40100

Sum of the first 300 terms = 40,100

List of multiples of 5 from 101 to 300

= 105,110,115,...,295,300

First term (a) = 105

Common Difference (d) = 110-105 = 5

Last term (l) = 300

We know that

l = a+(n-1)d

=> 300 = 105+(n-1)(5)

=> 300 = 105+5n-5

=> 300 = 100+5n

=> 300-100 = 5n

=> 200 = 5n

=> 5n = 200

=> n = 200/5

=> n = 40

Number of terms (n) = 40

We know that

Sum of the first 'n' terms in an AP

Sum of the first 'n' terms in an AP=> (n/2)[2a+(n-1)d]

On substituting these values in the above formula then

=> (40/2)[2(105)+(40-1)(5)]

=> 20[210+39(5)]

=> 20(210+195)

=> 20(405)

=> 8100

The sum of the multiples of 5 from 101 and 300 is 8100

The sum of all the non multiples of 5 from 101 and 300

= Sum of all numbers from 101 and 300 - Sum of all multiples of 5 from 101 and 300

= 40100-8100

= 32000

Answer :-

♦ The sum of all the non multiples of 5 from 101 and 300 = 32,000

Used Formulae:-

♦ Sum of the first 'n' terms in an AP

Sum of the first 'n' terms in an AP= (n/2)[2a+(n-1)d]

♦ Last term in an AP = l = a+(n-1)d

• a = First term
• d = Common Difference
• n = Number of terms
• l = last term