Geometric Series Worksheet Solution3





In the page geometric series worksheet solution3 you are going to see solution of each questions from the geometric series worksheet.

(6) The second term of the geometric series is 3 and the common ratio is 4/5. Find the sum of first 23 consecutive terms in the given geometric series.

Solution:

second term = 3

           t₂ = 3

         a r = 3    here r = 4/5 < 1   and n = 23

        a (4/5) = 3

           a = (3 x 5)/4

           a = 15/4 

Sum of first 23 terms = a (1-r^n)/(1-r)

                             = 15/4 (1-(4/5)^23)/(1-(4/5))

                             = 15/4 (1-(4/5)^23)/((5-4)/5)

                             = 15/4 (1-(4/5)^23)/(1/5)

                             = (15/4) x (5/1) (1-(4/5)^23)

                             = (75/4) (1-(4/5)^23)

                             = (75/4) (1-(4/5)^23)


(7) A geometric series consists of four terms and has a positive common ratio. The sum of the first two terms is 9 and the sum of the last two terms is 36. Find the series.

Solution:

 Let a,ar ,ar² and ar³ are the first four terms of the given geometric series

sum of the first two terms = 9

sum of the last two terms = 36

                          a + a r  = 9

                        a (1+ r) = 9

                       ar² + ar³ = 36

                      a r² (1 + r) = 36    --- (1)

Substitute a (1 + r) = 9 in the first equation

                      r² (9) = 36

                          r² = 36/9

                          r² = 4

                          r = √4

                          r = ± 2

    r = -2 is not admissible r = 2

                   a (1 + 2) = 9

                     a (3) = 9

                          a = 9/3

                          a = 3

so   3 + 3(2) + 3(2)² + 3(2)³+ .........

Therefore the series  is 3 + 6 + 12 + 24 + ......                                 


(8) Find the sum of the first n terms of the geometric series

(i) 7 + 77 + 777 + ..............

Solution:

  = [7 + 77 + 777 + ..............to  n terms]

  = 7 [1 + 11 + 111 + ..............to  n terms]

  = 7/9[9 + 99 + 999 + .............. to  n terms]

  = 7/9[(10-1) + (100-1) + (1000-1) + .............. to n terms]

  = 7/9[(10+100+1000+.............. to n terms)-(1+1+1+......to n terms)]

a = 10     r = 100/10

              r = 10 > 1

a = 1     r = 1/1

            r = 1 

   = 7/9 [10(10^n - 1)/(10-1) - n(1)]

   = 7/9 [10(10^n - 1)/9 - n]

   = 70/81(10^n - 1) - 7n/9

   = 70/81(10^n - 1) - n/9


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geometric series worksheet solution3