Finding the Middle Term of a Binomial Expansion

MathematicsBinomial TheoremMedium

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Question

Problem 4: Find the middle term of $(2p - \frac{1}{2q})^{10}$.

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Step by Step Written Solution

1
Step 1

Hi Theetso, let's find the middle term of this binomial expansion together.

Middle Term of Binomial Expansion

$$(2p - \frac{1}{2q})^{10}$$
2
Step 2

First, let's recall that if we have a power $n$, the total number of terms is $n$ plus one.

$$n = 10 \implies \text{Total terms} = 11$$
3
Step 3

Since we have eleven terms, the middle term is the sixth term.

4
Step 4

The general formula for the term $r$ plus one in a binomial expansion is $n$ choose $r$, times the first term to the power $n$ minus $r$, times the second term to the power $r$.

$$T_{r+1} = \binom{n}{r} a^{n-r} b^r$$
5
Step 5

Now, let's identify our variables for the sixth term. For $T_6$, $r$ must be equal to five.

Identifying Variables

$$n = 10$$
$$r = 5$$
$$a = 2p$$
$$b = -\frac{1}{2q}$$
6
Step 6

Let's substitute these values into our general term formula.

$$T_6 = \binom{10}{5} (2p)^{10-5} (-\frac{1}{2q})^5$$
7
Step 7

First, we simplify the exponents.

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About This Question

Subject
Mathematics
Topic
Binomial Theorem
Difficulty
Medium
Question Type
Open Ended

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