Sequence x1,x2,x3,...is satisfy x1+x2+x3+...+xn= n^3. What is the value of x100 ?

Question

DWRead · Accepted Answer

x₁ + x₂ + x₃ + ... + x₉₉ + x₁₀₀ = 100³
x₁ + x₂ + x₃ + ... + x₉₉ = 99³

Subtract the second equation from the first:
x₁₀₀ = 100³ - 99³ = 29,701

Prateek · Answer

It's easy.

x100
= (x1+x2+x3...+x100) - (x1+x2+x3...+x99)
= (100^3) - (99^3)
= 100^3 - (100-1)^3
= 100^3 - (100^3 - 1^3 - 3*100*1 (100-1))
= 100^3 - (100^3 - 1 - 300*99)
= 1 + 300*99
= 1 + 300*(100-1)
= 1 + 30000 - 300
= 1 + 29700
= 29701

Sequence x1,x2,x3,...is satisfy x1+x2+x3+...+xn= n^3. What is the value of x100 ?

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