A math problem for u guys cuz i can't solve it

Calculate S = [1 / √2 + √1] + [1 / √3 + √2] + … + [1 / √100 + √99].
Determine the smallest natural number n such that: [1 / √2 + √1] + [1 / √3 + √2] + … + [1 /√n+1 + √n].
1. We can do it with the calculator but it will take too much time. Is there a faster way to do it?
2.[1 / √2 + √1] + [1 / √3 + √2] + … + [1 / √n+1 + √n]
= (√2+√1) + (√3+√2) + … + (√n+1 + √n)
= √n+1 – 1
so finding the first n such that the sum in question exceeds 100 amounts to solving:
√n+1 – 1 ≥ 100
⇔ √n+1 ≥ 101
⇔ (√n+1)² ≥ 101²
⇔ n+1 ≥ 10201
⇔ n ≥ 10200
Is it good ?

2 years ago 49 views 1 frames 1 Like