Question

Evaluate:[(1/2)^-1 -(1/3)^-1]^-1

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Answer 1

What is the value of the product (1 - 1/2) (1 - 1/3) (1 - 1/4) (1 - 1/5)?

(1–1/2)=1/2

(1–1/2)(1–1/3)=1/2×2/3=1/3

(1–1/2)(1–1/3)(1–1/4)=1/2×2/3×3/4=1/3×3/4=1/4

(1–1/2)(1–1/3)(1–1/4)(1–1/5)=1/2×2/3×3/4×4/5=1/4×4/5=1/5

As you can see, the denominator of each term cancels out the numerator of the next term, so this series can be generalised to:

(1–1/2)(1–1/3)(1–1/4)…(1–1/n)=1/n

Which can also be written as:

1/2×2/3×3/4×…×((n-1)/n)=1/n

Which we could even be further generalised to:

a/b×b/c×c/d×…×m/n = a/n

hope it helps you ✌