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$What is the 4th term of the expansion of (1-2x)^(n) if the binomial coefficients are taken from the row of Pascal's triangle shown below? {:[1,6,15,20,15,6,1]:} 240x^(4) -20x^(3) -160x^(3) 160x^(3)$

What is the $4$ th term of the expansion of $(1-2 x)^{n}$ if the binomial coefficients are taken from the row of Pascal's triangle shown below? $\newline$ $\begin{array}{lllllll} 1 & 6 & 15 & 20 & 15 & 6 & 1 \end{array}$ $\newline$ $240 x^{4}$ $\newline$ $-20 x^{3}$ $\newline$ $-160 x^{3}$ $\newline$ $160 x^{3}$

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Math Problems

Precalculus

Pascal's triangle and the Binomial Theorem

Full solution

Q. What is the

4

th term of the expansion of

(1-2 x)^{n}

if the binomial coefficients are taken from the row of Pascal's triangle shown below?

\newline

\begin{array}{lllllll} 1 & 6 & 15 & 20 & 15 & 6 & 1 \end{array}

\newline

240 x^{4}

\newline

-20 x^{3}

\newline

-160 x^{3}

\newline

160 x^{3}

Identify general term: Identify the general term in the binomial expansion, which is given by $T(k+1) = C(n, k) \cdot a^{(n-k)} \cdot b^k$ , where $C(n, k)$ is the binomial coefficient, $a$ and $b$ are the terms in the binomial, and $k$ is the term number
Calculate $4$ th term: For the $4$ th term $k=3$ , use the binomial coefficient $C(n, 3) = 20$ from Pascal's triangle. The terms in the binomial are $a = 1$ and $b = -2x$ . Calculate the $4$ th term: $T(4) = 20 \times 1^{n-3} \times (-2x)^3$
Simplify expression: Simplify the expression: $T(4) = 20 \times 1 \times (-8x^3) = -160x^3$