Application of the Binomial Theorem
Read this topicProblems 11
- Find the sum of all positive integers 1≤k≤991\leq k\leq 991≤k≤99 such that there exist positive integers aaa and bbb with the property that x100−axk+b=(x2−2x+1)P(x)x^{100}-ax^k+…
- Find the coefficient of the x4yx^4yx4y term in the expansion of (3x+4y)5(3x+4y)^5(3x+4y)5.
- What is the value of 2552255^22552? Write your answer without commas or spaces.
- Compute 8652.865^2.8652.
- Find the coefficient of a1/2a^{1/2}a1/2 in the expansion of (a+1a)5\left( a + \frac{1}{\sqrt a} \right)^5(a+a1)5.
- What is the coefficient of a−3a^{-3}a−3 in the expansion of (1−1a)7\left( 1 - \frac1a \right)^7(1−a1)7?
- When the integer (3+5)103−(3−5)103{\left(\sqrt{3} + 5\right)}^{103} - {\left(\sqrt{3} - 5\right)}^{103}(3+5)103−(3−5)103 is divided by 9, what is the remainder? Source: Math P…
- The number 104,060,465104{,}060{,}465104,060,465 is divisible by a five-digit prime number. What is that prime number? (Hint: Do the digits look like a row of Pascal's triangle?…
- What is the full expansion of (x−3)4(x-3)^4(x−3)4?
- What is the full expansion of (x+3)3(x+3)^3(x+3)3?
- Use the binomial theorem to estimate (1.05)4(1.05)^4(1.05)4 to the tenths digit.