Special Series Worksheet Solution2





In this page special series worksheet solution2 we are going to see solution for each problems in the worksheet.

(5) Find the sum of the special series 5² + + 9² + ......... + 39²

Solution:

To find sum of this series we have to apply the formula for sum of squares of n natural numbers.But this sequence starts from 5 and we have only odd numbers. To make this as normal form we have to find the sum of squares of all numbers starts from 1 to 39. This is containing sum of odd and even numbers.Here we don't want sum of even numbers so we have to subtract their squares from the whole.

Sum of squares of n natural numbers = n(n+1)(2n+1)/6

n represents the number of terms in the given series


        = (1²+2²+3²+......... +39²)-(2²+4²+6²+......... +38²)-(1²+3²)

        = (1²+2²+3²+......... +39²)-2²(1²+2²+3²+......... +19²)-(1+9)

        = (1²+2²+3²+......... +39²)-4(1²+2²+3²+......... +19²)-10

So,     = [39 (39+1) (2x39+1)]/6 - 4[19 (19+1) (2x19+1)]/6 - 10

         = [39 x 40 x 79]/6 - 4[19 x 20 x 39]/6 - 10

         = 20540 - 4 (2470) - 10

         = 20540 - 9880 - 10

         = 20540 - 9890

         = 10650


(6) Find the sum of the special series 16³ + 17³ + ......... + 3

Solution:

To find sum of this series we have to apply the formula for sum of cubes of n natural numbers.But this sequence starts from 16. So we have to find sum of cubes from 1 up to 35 and subtract the sum of cubes of series from 1 up to 15 from this.

Sum of cubes of n natural numbers = [n(n+1)/2]²

        = ( 1³ + 2³ + 3³ + ......... +35³ )-( 1³ + 2³ + 3³ +......... + 15³ )

        =  [35(35+1)/2]² - [15(15+1)/2]²

        =  [(35 x 36)/2]² - [(15 x 16)/2]²

        =  [35 x 18]² - [15 x 8]²

        =  630² - 120²

        =  396900 - 14400

        = 382500


(7) Find the value of k if 1³ + 2³ + 3³ + ......... + k³ = 6084

Solution:

Here 6084 represents sum of cubes of natural numbers up to k. So we can write the formula instead of the series 1³ + 2³ + 3³ + ......... + k³.

Sum of cubes of n natural numbers = [n(n+1)/2]²

        [k(k+1)/2]² = 6084

        [k(k+1)/2] = √6084

        [k(k+1)/2] = √78 x 78

        [k(k+1)/2] = 78

        (k²+ k)/2 = 78

          k²+ k = 78 x 2

          k²+ k = 156

          k²+ k - 156 = 0

        (k - 12) (k+13) = 0

        k - 12 = 0         k + 13 = 0

              k = 12             k = -13   

k = -13 is not admissible. Therefore k = 12 is the solution.



These are the contents in the page special series worksheet solution2.

special series worksheet solution2