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.

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多项填空题

类似问题

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)

The implementation and loading phase of the Database Life Cycle (DBLC) involves _____.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单