The first three terms in the binomial expansion of (a+b)n are given to be 729,7209, and 30375 respectively. Find a,b, and n.
Here, first 3 terms are an,nC1an−1b and nC2an−2b2.
Also, it is given that their values are 729,7290 and 30375.
Therefore,
an=729 …… (1)
nC1an−1b=7290 …… (2)
nC2an−2b2=30375 …… (3)
Divide (1) by (2).
annC1an−1b=7297290
annan−1b=110
anb=110 …… (4)
Divide (2) by (3).
nC1an−1bnC2an−2b2=729030375
2(n−1)×an−1−(n−2)1×bb2=625
2a(n−1)b=625 …… (5)
Now, divide (4) by (5).
anb2a(n−1)b=(110)(625)
n−12n=512
12n−12=10n
n=6
Therefore,
a6=729
a=3
And,
2a(n−1)b=625
5b=25
b=5
Hence,
a=3,b=5 and n=6