Question

Geetha applies the Binomial Theorem for the expansion of (3x−5y)
5
. Which of the following is the last term in the expression?
A. −3,125y
5

B. 3.125⋅4
5

C. 5y
3

D. −5y
5

Answer 1

To determine the last term in the expansion of \((3x - 5y)^5\) using the Binomial Theorem, we can express the expansion as follows:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In our case, \(a = 3x\), \(b = -5y\), and \(n = 5\). According to the theorem, the last term occurs when \(k = n\), i.e., \(k=5\).

The general term is given by:

\[
T_k = \binom{n}{k} a^{n-k} b^k
\]

For our specific case, the last term (\(k=5\)) is:

\[
T_5 = \binom{5}{5} (3x)^{5-5} (-5y)^5
\]

Calculating this step by step:

1. \(\binom{5}{5} = 1\)
2. \((3x)^{0} = 1\) (since any number to the power of zero is 1)
3. \((-5y)^{5} = -5^5 y^5 = -3125y^5\) (since \(-5^5 = -3125\))

Putting it all together:

\[
T_5 = 1 \cdot 1 \cdot (-3125y^5) = -3125y^5
\]

Thus, the last term in the expansion of \((3x - 5y)^5\) is:

\[
\text{Last term} = -3125y^5
\]

So, the correct answer is:

**A. −3,125y^5**