A radical sum with one extra endpoint

Given ∑r=1250014r−3+4r+1=10001−14\sum_{r=1}^{2500}\frac1{\sqrt{4r-3}+\sqrt{4r+1}}=\frac{\sqrt{10001}-1}4, hence show, without a calculator, that ∑r=0250014r+1+4r+5>24\sum_{r=0}^{2500}\frac1{\sqrt{4r+1}+\sqrt{4r+5}}>24. marks

[modified 2012/ACJC/Prelim/I/10]

Solution

Shift the index

Match the given sum

Given ∑r=1250014r−3+4r+1=10001−14\sum_{r=1}^{2500}\frac1{\sqrt{4r-3}+\sqrt{4r+1}}=\frac{\sqrt{10001}-1}4, express ∑r=0250014r+1+4r+5\sum_{r=0}^{2500}\frac1{\sqrt{4r+1}+\sqrt{4r+5}} in terms of it.

Explore this idea

Hint

  1. Replace rr by r−1r-1 so the radicals match.

  2. Adjust the limits, then split off the extra term.

Answer · Shift the index

The sum equals 10001−14+110001+10005\dfrac{\sqrt{10001}-1}4+\dfrac1{\sqrt{10001}+\sqrt{10005}}.

Explore the idea

Radical terms that telescope

Rationalising turns each term into 4r+5−4r+14\frac{\sqrt{4r+5}-\sqrt{4r+1}}4, so the sum from 00 to NN is 4N+5−14\frac{\sqrt{4N+5}-1}4.

Upper limitr = 0 to 5
Try an example
Σ 1/(√(4r+1) + √(4r+5)), r = 0 to 5
1
Number of terms
6
(√(4N+5) − 1)/4
1
matches the direct total
Key idea

Each term is a difference of square roots, so the sum collapses to the last root minus the first.

Ideas to try
  1. At N=0N=0 the single term is 11+5\frac1{1+\sqrt5}, and the closed form gives 5−14\frac{\sqrt5-1}4. These are equal, since 11+5=5−15−1\frac1{1+\sqrt5}=\frac{\sqrt5-1}{5-1}.
  2. Slide NN upward: the direct total and the closed form agree every time.
Show it exceeds 24

Bound a root without a calculator

Hence show, without a calculator, that ∑r=0250014r+1+4r+5>24\sum_{r=0}^{2500}\frac1{\sqrt{4r+1}+\sqrt{4r+5}}>24.

Hint

  1. Compare 10001\sqrt{10001} with 10000=100\sqrt{10000}=100.

  2. Drop the positive extra term.

Answer · Show it exceeds 24

The sum is (10001−1)/4+1/(10001+10005)>99/4>24(\sqrt{10001}-1)/4+1/(\sqrt{10001}+\sqrt{10005})>99/4>24.

Takeaway

Rationalise to telescope

Radical denominators of the form a+b\sqrt a+\sqrt b rationalise to b−ab−a\frac{\sqrt b-\sqrt a}{b-a}, a difference that cancels between neighbours.