## Special Series Worksheet Solution3

In this page special series worksheet solution3 we are going to see solution for each problems in the worksheet.

(8) Find the value of p if 1³ + 2³ + 3³ + ......... + p³ = 2025

Solution:

Here 2025 represents sum of cubes of natural numbers up to p. So we can write the formula instead of the series 1³ + 2³ + 3³ + ......... + p³.

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

[p(p+1)/2]² = 2025

[p(p+1)/2] = √2025

[p(p+1)/2] = √45 x 45

[p(p+1)/2] = 45

(p²+ p)/2 = 45

p²+ p = 45 x 2

p²+ p = 90

p²+ p - 90 = 0

(p - 9) (p + 10) = 0

p - 9 = 0         p + 10 = 0

p = 9             p = -10

p = -10 is not admissible. Therefore p = 9 is the solution.

(9) If  1 + 2 + 3 + ....... + p = 171 then find 1³ + 2³ + 3³ + ......... + p³

Solution:

Here 171 represents sum of natural numbers up to p.From this we have to find sum of cubes of natural numbers up to p.

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

1 + 2 + 3 + ....... + p = 171

Sum of natural numbers up to p = 171

p ( p + 1 )/2 = 171

1³ + 2³ + 3³ + ......... + p³  = (171)²

1³ + 2³ + 3³ + ......... + p³  = 171 x 171

1³ + 2³ + 3³ + ......... + p³  = 29241

(10) If  1³ + 2³ + 3³ + ......... + k³  = 8281 then find 1 + 2 + 3 + .........+k

Solution:

Here 8281 represents sum of cubes of natural numbers up to k.From this we have to find sum of natural numbers up to k.

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

1³ + 2³ + 3³ + ......... + k³  = 8281

Sum of cubes natural numbers up to k = 8281

[k ( k + 1 )/2]² = 8281

[k ( k + 1 )/2] = √8281

[k ( k + 1 )/2] = √91 x 91

[k ( k + 1 )/2] = 91

Sum of natural numbers up to k = 91

These are the contents in the page special series worksheet solution3.