Progressions

Algebra Level 4

48 2 × 3 + 47 3 × 4 + 46 4 × 5 + ⋯ + 1 49 × 50 \dfrac{48}{2\times3} + \dfrac{47}{3\times4} + \dfrac{46}{4\times5} + \cdots + \dfrac{1}{49\times50}

If the value of the expression above is also equal to

51 2 + k ( 1 + 1 2 + 1 3 + ⋯ + 1 50 ) \dfrac{51}2 + k \left( 1 + \dfrac12 + \dfrac13 + \cdots + \dfrac1{50} \right)

for some constant k k , find k k .


The answer is -1.

Vibhu Agarwal by Vibhu Agarwal , 25, India

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1 solution

Pedro Saboya
Aug 28, 2020

S = 48(1/2-1/3) + 47(1/3-14) + ... + 1(1/49-1/50) = 48/2 - (1/3+1/4+...+1/49+1/50) = 51/2 - (1/1+1/2+...+1/50)

Logo, k = -1.

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