FIND THE SUM OF ARITHMETIC SERIES WITH GIVEN DESCRIPTION

Problem 1 :

Find the sum of all 3 digit natural numbers, which are divisible by 9.

Solution :

3 digit number starts from 100 and ends with 999. From this sequence we have to find number of terms which are divisible by 9 and also we have to find their sum.

  108, 117, 126,........................999

The first number which is divisible by 9 from this sequence is 108, the second number will be 117 and the last number

will be 999.

108 + 117+ 126 + ........... + 999

a = 108, d = 117 - 108 = 9 and l = 999      

to find number of terms, we have to use the formula for (n)

  n  =  [(l-a)/d]+1

  n  =  [(999-108)/9] + 1

  n  =  [891/9] + 1

  n  =  99 + 1

  n  =  100

 S_n  =  (n/2)[a+l]

= (100/2)[108+999]

=  50 (1107)

=  55350

Problem 2 :

Find the sum of first 20 terms of the arithmetic series in which 3^rd term is 7 and 7^th term is 2 more than three times its 3^rd term.

Solution :

t₃  =  7

t₇  =  3t₃+2

t₇  =  3(7)+2

t₇  =  23

a+2d  =  7   -----(1)

a+6d  =  23 -----(2)

(1) - (2)

-4d  =  -16

d  =  4

By applying the value of d in (1), we get

a = -1

Now we need to find sum of 20 terms

S_n  =  (n/2)[2a+(n-1)d]

S₂₀  =  (20/2) [2(-1)+(20-1)(4) ]

=  10 [-2+19(4)]

=  10 (-2+76)

=  740

Problem 3 :

In an arithmetic series, the sum of first 11 terms is 44 and the that of the next 11 terms is 55.

Find the arithmetic series.

Solution :

Sum of first 11 terms  =  44

S₁₁  =  44

(11/2)[2a+(11-1)d]  =  44

  2a+10d  =  44 ⋅ (2/11)

  2a+10d  =  8   ----- (1)

Sum of next 11 term  =  55

  S₂₂  =  S₁₁ + 55

S₂₂  =  44 + 55

S₂₂  =  99

(22/2)[2a+(22-1)d]  =  99

2a+21d  =  99/11

2a+21d  =  9   ----- (2)

(1) - (2)

-11d  =  -1

  d = 1/11

By applying the value of d = 1/11 in (1), we get

  2a+10(1/11)  =  8

  2a+10/11  =  8

  2a  =  8-(10/11) ==> 78/11 ==> a = 39/11

Therefore the series is

(39/11) + (40/11) + (41/11) + ............

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.