Algebra
Binomial Theorem
MMTS_Full_Test_11
Grade 12
Question:
If $(1+x)^{50}=a_0+a_1x+\cdots+a_{50}x^{50}$, then $\dfrac{a_1+a_2+\cdots+a_{25}}{a_{26}+a_{27}+\cdots+a_{50}}$ equals
Step-by-Step Solution
Key Concept: Symmetry of binomial coefficients: $a_k=\binom{50}{k}$
$a_k=\binom{50}{k}=a_{50-k}$. $a_{25}=\binom{50}{25}$. $\sum_{k=1}^{25}a_k=\sum_{k=25}^{49}a_k$... by symmetry numerator$=$ denominator. Ratio $=1$.
Correct Answer: 2