Sum of Cubes of First n Natural Numbers

Example 1 :

Find the value of

1³ + 2³ + 3³ + ........ + 17³

Solution :

= 1³ + 2³ + 3³ + ........ + 17³

Using 1³ + 2³ + 3³ + ........ + n³ = [n(n + 1)/2]²,

= [17(17 + 1)/2]²

= [17(18)/2]²

= [17(9)]²

= 153²

= 23409

Example 2 : 

Find the value of

8³ + 10³ + 11³ + ........ + 21³

Solution :

8³ + 10³ + 11³ + ........ + 21³ :

= (1³ + 2³ + 3³ + ........ + 21³) - (1³ + 2³ + 3³ + ........ + 7³)

Using 1³ + 2³ + 3³ + ........ + n³ = [n(n + 1)/2]²,

= [21(21 + 1)/2]² - [7(7 + 1)/2]²

= [21(22)/2]² - [7(8)/2]²

= [21(11)]² - [7(4)]²

= 231² - 28²

= 53361 - 784

= 52577

Example 3 : 

Find the value of

3³ + 6³ + 9³ + ........ + 45³

Solution :

= 3³ + 6³ + 9³ + ........ + 45³

= (3x1)³ + (3x2)³ + (3x3)³ + ........ + (3x15)³

= 3³1³ + 3³2³ + 3³3³ + ........ + 3³15³

= 3³(1³ + 2³ + 3³ + ........ + 15³)

Using 1³ + 2³ + 3³ + ........ + n³ = [n(n + 1)/2]²,

= 3³[15(15 + 1)/2]²

= 3³[15(16)/2]²

= 3³[15(8)]²

= 27(120)²

= 27(14400)

= 388800

Example 4 : 

Find the sum of 

2 + 16 + 54 + ........ + to 12 terms

Solution :

2 + 16 + 54 + ........ + to 12 terms

=  2(1 + 8 + 27 + ........ to 12 terms)

=  2(1³ + 2³ + 3³ + ........ to 12 terms)

=  2(1² + 2² + 3² + ........ + 12²)

Using 1³ + 2³ + 3³ + ........ + n³ = [n(n + 1)/2]²,

= 2[12(12 + 1)/2]²

= 2[12(13)/2]²

= 2[6(13)]²

= 2(78)²

= 2(6084)

= 12164

Example 5 : 

Find the sum of

3 + 24 + 81 + ........ + 6591

Solution :

3 + 24 + 81 + ........ + 6591 :

= 3(1 + 8 + 27 + ........ + 2197)

=  3(1³ + 2³ + 3³ + ........ + 13³)

Using 1³ + 2³ + 3³ + ........ + n³ = [n(n + 1)/2]²,

= 3[13(13 + 1)/2]²

= 3[13(14)/2]²

= 3[6(7)]²

= 3(42)²

= 3(1764)

= 5292

Example 6 :

Find the average of cubes first 10 natural numbers. 

Solution :

Using 1³ + 2³ + 3³ + ........ + n³ = [n(n + 1)/2]², to find the sum of squares first 10 natural numbers. 

1³ + 2³ + 3³ + ........ + 10³ :

= [10(10 + 1)/2]²

= [10(11)/2]²

= [5(11)]²

= 55²

= 3025

Average of cubes first 10 natural numbers : 

= (Sum of cubes first 10 natural numbers)/10

= 3025/10

= 302.5

Example 7 : 

If 1 + 2 + 3 + ........ + k = 325, then find 

1³ + 2³ + 3³ + ........ + k³

Solution :

1 + 2 + 3 + ........ + k = 325

Using 1 + 2 + 3 + ........ + n = n(n + 1)/2, 

k(k + 1)/2 = 325

Using 1³ + 2³ + 3³ + ........ + n³ = [n(n + 1)/2]², 

1³ + 2³ + 3³ + ........ + k³ = [k(k + 1)/2]²

Substitute k(k + 1)/2 = 325, 

1³ + 2³ + 3³ + ........ + k³ = 325²

1³ + 2³ + 3³ + ........ + k³ = 105625

Example 8 : 

If 1³ + 2³ + 3³ + ........ + k³ = 44100, then find 

1 + 2 + 3 + ........ + k

Solution :

1³ + 2³ + 3³ + ........ + k³ = 44100

Using 1³ + 2³ + 3³ + ........ + n³ = [n(n + 1)/2]²,

[k(k + 1)/2]² = 44100

[k(k + 1)/2]² = 210²

k(k + 1)/2 = 210

Using 1 + 2 + 3 + ........ + n = n(n + 1)/2, 

1 + 2 + 3 + ........ + k = 210

Example 9 : 

How many terms of the series 1³ + 2³ + 3³ + ........ should be taken to get the sum 14400?

Solution :

Let n be the number of terms to be taken to get the sum 14400.

1³ + 2³ + 3³ + ........ to n terms = 14400

1³ + 2³ + 3³ + ........ + n³ = 14400

[n(n + 1)/2]² = 14400

[n(n + 1)/2]² = 120²

n(n + 1)/2 = 120

Multiply each side by 2.

n(n + 1) = 240

n² + n = 240

n² + n - 240 = 0

Factor and solve. 

n² + 16n - 15n - 240 = 0

n(n + 16) - 15(n + 16) = 0

(n + 16)(n - 15) = 0 

But n ≠ -16, because number of terms can not be a negative value.

Hence n = 15.

In the given series, 15 terms to be taken to get the sum 14400.

Example 10 :

The sum of the squares of the first n natural numbers is 285, while the sum of their cubes is 2025. Find the value of n.

Solution :

Sum of the squares of first n natural numbers is 285 :

1² + 2² + 3² + ........ + n² = 285

[n(n + 1)(2n + 1)]/6 = 285

Multiply each side by 6.

n(n + 1)(2n + 1) = 1710 ----(1)

Sum of their cubes is 2025 :

1³ + 2³ + 3³ + ........  + n³ = 2025

[n(n + 1)/2]² = 2025

[n(n + 1)/2]² = 45²

n(n + 1)/2 = 45

Multiply each side by 2.

n(n + 1) = 90

Substitute n(n + 10 = 90 in (1). 

90(2n + 1) = 1710

Divide each side by 90.

2n + 1 = 19

Subtract 1 from each side. 

2n = 18

Divide each side by 2. 

n = 9

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