Question

2. Find the value of k if
(i) 1 + 2 + 3 +......+ k = 6084 (ii) 1 + 2 + 3 +.....+ k = 2025

Answer 1

QUESTION:
Find the value of k if 1³ + 2³ + 3³ + ......... + k³ = 6084
Find the value of k if 1³ + 2³ + 3³ + ......... + k³ = 2025

SOLUTION :
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
k²- 12k +13k -156
[By middle term splitting]
k(k -12) +13(k -12)= 0
(k - 12) (k+13) = 0
k - 12 = 0 or k + 13 = 0
k = 12 or k = -13 [neglected]

Hence, k = 12 is the solution.

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

[k(k+1)/2]² = 2025
[k(k+1)/2] = √2025
[k(k+1)/2] = √45 x 45
[k(k+1)/2] = 45
(k²+ k)/2 = 45
k²+ k = 45 x 2
k²+ k= 90
k²+ k- 90 = 0
[By middle term splitting]
k²-9k + 10k -90 = 0
k(k -9) +10(k -9)= 0
(k - 9) (k + 10) = 0
k - 9 = 0 or k + 10 = 0
k = 9 or k = -10 [Neglected]

Hence, k = 9 is the solution.

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