Question

How do you find the last two terms in the expansion of (a^2/3 + a^1/3)^25?

Answer 1

here's the equation: \[(a ^{\frac{ 2 }{ 3 }} + a ^{\frac{ 1 }{ 3 }})^{25}\]

Answer 2

\[(x+y)^n=\sum_\left\{ k=1 \right\}^n \left(\begin{matrix}n \\ k\end{matrix}\right) x ^\left\{ n-k \right\} y^k\]

Answer 3

write the equation with square roots

Answer 4

The last two terms are at k=24 and k=25, if you know how to simplify the binomial coefficient that will help you a lot too.

Answer 5

@Spacelimbus i know the last 2 terms are 24 and 25. just don't know what to do after...

Answer 6

Use the binomial theorem

\[\Large (x+y)^n=\sum_\left\{ k=1 \right\}^n \left(\begin{matrix}n \\ k\end{matrix}\right) x ^\left\{ n-k \right\} y^k\]
You want the last two, so k=24 and then k = 25 for the second last and last terms
n=25
x = (a^2/3)
y = (a^1/3)

\[\Large 25C24 (a ^{\frac{ 2 }{ 3 }})^{25-24} (a ^{\frac{ 1 }{ 3 }})^{24} \]

that's the 2nd last term...simplify and try to find the last.

Answer 7

And as a small help

\[25C24=25C(25-24)=25C1=25\]

Answer 8

@agent0smith and @Spacelimbus thanks for the help! :)

Answer 9

No prob, if you need any help getting the very last term, let me know :)

Answer 10

btw note that the last two terms are actually the 25th and 26th terms.

The term is always one more than the k value.

Eg. If you're asked to find the 10th term, that means k=9 in the binomial formula. If you need the 4th term, k=3.

This is because the 1st term is when k=0.