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 **5****² + 7² + 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 + ......... + 35³**

**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.

**Special series worksheet****Special series****Sequence****Arithmetic progression****Arithmetic series****Geometric progression****Geometric series**

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special series worksheet solution2