Question textConsider the following sequence for which three terms are shown:[math: p=3!1!+5!3!+7!5!+…]In general: [math: pi=(2i+1)!(2i−1)!=(2i)(2i+1)]What is the minimum number of terms required for p to be greater than 131,313? Answer 1 Question 2[input]Check Question 2

Question

BlackTom AI · Accepted Answer

Each term simplifies because (2i+1)!/(2i−1)! = (2i)(2i+1). Summing the first n terms gives
p(n) = sum_{i=1}^n [4i^2 + 2i] = 4 sum_{i=1}^n i^2 + 2 sum_{i=1}^n i.

Use standard formulas: sum i = n(n+1)/2 and sum i^2 = n(n+1)(2n+1)/6. Substitute:
p(n) = 4·[n(n+1)(2n+1)/6] + 2·[n(n+1)/2]
= (2/3)n(n+1)(2n+1) + n(n+1)
Factor n(n+1): p(n) = n(n+1)[(2/3)(2n+1) + 1]
Combine terms: (2/3)(2n+1)+1 = (4n+2)/3 + 1 = (4n+5)/3
So closed form: p(n) = n(n+1)(4n+5)/3.

We need the smallest integer n with p(n) > 131,313. Multiply both sides by 3:
n(n+1)(4n+5) > 393,939.

Check n = 45:
p(45) = 45·46·(4·45+5)/3 = 45·46·185/3.
45/3 = 15, so p(45) = 15·46·185.
46·185 = 8510, so p(45) = 15·8510 = 127,650, which is less than 131,313.

Check n = 46:
p(46) = 46·47·(4·46+5)/3 = 46·47·189/3.
189/3 = 63, so p(46) = 46·47·63.
46·47 = 2,162 and 2,162·63 = 136,206, which is greater than 131,313.

Thus n = 45 is insufficient while n = 46 meets the requirement; there is no smaller n > 45 that would exceed the threshold. Therefore the minimum number of terms required is 46.

Final answer: 46.

Similar Questions

What is the sum of Series 2?

Match the following: 1: 2 + 6 + 18 + ... 2: 5, 11, 17, ... 3: 4, 12, 36, .... 4: 3 + 7 + 11 + ....

Continued from the previous question.How many days before 20 000 litres of water in total has seeped into the cave? (write the value only, units not required)

设 𝑎 𝑛 = 𝑛 2 − 53 𝑛 + 196 + ∣ 𝑛 2 − 53 𝑛 + 196 ∣ , 则 ∑ 𝑛 = 0 50 𝑎 𝑛 = ( ) Let 𝑎 𝑛 = 𝑛 2 − 53 𝑛 + 196 + ∣ 𝑛 2 − 53 𝑛 + 196 ∣ ，then ∑ 𝑛 = 0 50 𝑎 𝑛 = ( )

Continued from the previous question.How many litres of water in total has seeped into the cave after five days? (write the value only, units not required)

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler