HOW TO FIND THE SUM OF TERMS WHICH ARE MULTIPLES BETWEEN TWO NUMBERS

We can use the formula for sum of the terms in arithmetic sequence to find the sum of terms which are multiples between two numbers.

To find the sum of terms of in arithmetic sequence, we can use the formula given below.

Sn  =  (n/2) [a + l] (or)

Sn  =  (n/2) [2a + (n - 1)d]

Example 1 :

If the sum of the first n terms of an AP is 4n - n2  what is the first term (that is S)? what is the sum of first two terms?what is the second term? similarly find the 3rd,the 10th and the nth terms.

Solution :

Sn  =  4n - n2

Sum of first two terms  =  4

n = 1

   = 4 (1) - (1)2

    4 - 1 = 3

first term = 3

S n = 4 n - n2

n = 2

   = 4 (2) - (2)2

    8 - 4 = 4

To find the second term, we have to subtract the sum of two terms from first term

  =  4 - 3  =  1

Second term  =  1

a =  a + 2d

d = second term - first term  =  1 - 3  =  -2 

 a3 =  3 + 2(-2)  =  3 - 4 = -1

 a10  =  a + 9 d

=  3 + 9(-2)

= 3 - 18

= -15

Example 2 :

Find the sum of first 40 positive integers divisible by 6.

Solution :

The positive integers which are divisible by 6 are 

6, 12, 18,...........

6 + 12 + 18 + ...........

a = 6, d = 12 - 6  =  6, n = 40

Sn  =  (n/2) [ 2 a + (n - 1) d]

S40  =  (40/2) [2(6) + (40-1) (6)]

=  20 [12 + 39(6)]

=  20 [12 + 234]

=  20 [246]

=  4920

Example 3 :

Find the sum of first 15 multiples of 8.

Solution :

first 15 multiples of 8 are

8, 16, 24,................

8 + 16 + 24 + ...........

a = 8         d = 16 - 8       n = 15

                   = 8

 Sn  =   (n/2) [ 2 a + (n - 1) d]

 S15 = (15/2) [2(8) + (15-1) (8)]

  =  (15/2) [16 + 14(8)]

  =  (15/2) [16 + 112]

  =  (15/2) [128]

  =  15 [64]

  =  960

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