In the expansion of (5+x^2)^n in ascending powers of x, where n is a positive integer, the coefficient of the third term is 9375. Find the value of n.
I sub the coefficient into 9375. I dont know how separate n from 5^n-2...
Or did I use the wrong way to solve???
Please somebody help me~~~
F.4 amaths - binomial question
2006-11-02 4:43 am
回答 (3)
2006-11-02 6:13 am
✔ 最佳答案
In the expansion of (5+x2)n in ascending powers of x, where n is a positive integer, the coefficient of the third term is 9375. Find the value of n.5n + nC1(5n-1)x2 + nC25n-2x4 + ……
nC25n-2 = 9375
因9375 = 55x3
所以 n-2 ≦ 5，所以n ≦ 7
但 7C2 = 21
6C2 = 15
5C2 = 10
4C2 = 6
因沒 9375 只有3及5 的因子，所以只有n = 6為一可能
6C256-2 = 9375
所以 n = 6
2006-11-02 6:22 am
(5+x^2)^n = 5^n + nC1*5^(n-1)*x^2 + nC2*5^(n-2)*x^4 + ......
The coefficient of the 3rd term
= nC2*5^(n-2)
= [n(n-1)/2]*5^(n-2)
= n(n-1)*5^n/50
n(n-1)*5^n/50 = 9375
n(n-1)*5^n = 468750 = 2*3*5^7 = 6(6-1)*5^6
So n = 6
The coefficient of the 3rd term
= nC2*5^(n-2)
= [n(n-1)/2]*5^(n-2)
= n(n-1)*5^n/50
n(n-1)*5^n/50 = 9375
n(n-1)*5^n = 468750 = 2*3*5^7 = 6(6-1)*5^6
So n = 6
2006-11-02 4:51 am
simply by using logarithm
for example, A^(n-2)=B
then (n-2)logA=logB
for example, A^(n-2)=B
then (n-2)logA=logB
收錄日期: 2021-04-12 22:42:32
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https://hk.answers.yahoo.com/question/index?qid=20061101000051KK03915