Arithmetic Series Worksheet Solution6





In the page arithmetic series worksheet solution6 you are going to see solution of each questions from the arithmetic series worksheet.

(15) The sum of first n terms of a certain series is given as 3n² - 2n.Show that the series is an arithmetic series.

Solution:

Sn = 3n² - 2n

S₁ = 3(1)² - 2(1)

    = 3 - 2

S₁ = 1

a = 1

substitute n = 2 to get sum of first two terms

S₂ = 3(2)² - 2(2)

     = 3(4) - 4

     = 12 - 4

     = 8

a + a + d = 8

2a + d = 8

2(1) + d = 8

         d = 8 - 2 

         d = 6

let us find the sum of (n-1) terms using Sn

S (n-1) = 3 (n-1)² - 2 (n-1)

tn = Sn-S(n-1) 

    = 3n² - 2 n - [3(n-1)² - 2(n-1)]

tn = 3n² - 2 n - [3(n² + 1 - 2n) - 2(n-1)]

    = 3n² - 2 n - [3n² + 3 - 6n - 2n + 2]

    = 3n² - 2 n - 3n² - 3 + 6n + 2n - 2

    = 3n² - 2 n - 3n² - 3 + 6n + 2n - 2

    = 6n-5

    = 6 n - 6 + 1

    = 6 (n-1) + 1

    = 1 + (n-1) 6

    a + (n-1) d

by comparing with this the value of a is 1 and the value of d is 6.This value exactly matches with previous one. So that we can say the given forms an A.P.


(16) If a clock strikes once at 1'o clock,twice at 2'o clock and so on.How many times will it strike a day?

Solution:

The clock strikes once at 1'o clock,twice at 2'o clock and so on.Now we have to write this pattern as series because we need to find number of times that the clock strike a day.

           = 2 [1 + 2 + 3 + ............ + 12]

           = 2 x (n/2) [a+L]

           = 2 x (12/2) [1+12]

           = 12 [13]

           = 156

Therefore the clock will strike 156 times in a day.


(17) Show that the sum of an arithmetic series whose first term is a,second term is b and the last term is c is equal to [(a+c) (b+c-2a)]/2(b-a)

Solution:

From this series we know that the first term is a ,the second term is b and the last term is c

a = a     d = b-a  and L= c

Now we have to to find the value of n

          so   n = [(l-a)/d] + 1

                   = [(c-a)/(b-a)] + 1

                   = (c-a + b-a)/(b-a)

                   = (c + b - 2a)/(b-a)

Now we have to use the formula for Sn to find sum of n terms

               Sn =(n/2) [a+L]

                  = {[(c + b - 2a)/(b-a)]/2}[a+c]

                  = [(c + b - 2a)(a+c)]/2(b-a)                



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arithmetic series worksheet solution6