Binomial Theorem
Properties of Binomial Expansion
Premium Question
Grade 11
Question:
If $T_r$ denotes the $r^{\text{th}}$ term in the expansion of $(1 + x)^n$, then $\frac{T_{r+1}}{T_r}$ equals:
(1) $\frac{n - r + 1}{r} x$
(2) $\frac{n - r}{r + 1} x$
(3) $\frac{r}{n - r + 1} x$
(4) $\frac{n + r}{r} x$
Step-by-Step Solution
Key Concept: Write out $T_r = \binom{n}{r-1} x^{r-1}$ and $T_{r+1} = \binom{n}{r} x^r$ explicitly, form their ratio, and simplify $\frac{\binom{n}{r}}{\binom{n}{r-1}}$ using the factorial definition of the binomial coefficient.
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Correct Answer: (1)