solving for positive and negitive numbers

Indices - We know that 64 = 2 x 2 x 2 x 2 x 2 x 2 = 2 6 We know that 64 = 2 x 2 x 2 x 2 x 2 x 2 = 2 6 Here 2 is called the base and 6 is called the power (or index or exponent). We say that "64 is equal to base 2 raised to the power 6". Similarly, if m is a positive integer and then a a a..

Middle Terms for Positive Integral Index - The number of terms in the expansion of (a + b) n depends on the index n. The index n is either even or o..

The number of terms in the expansion of (a + b) n depends on the index n. The index n is either even or od..

Greatest Terms for Positive Integral Index - In (a + b) n , let 'a' and 'b' be both positive numbers. As r increases, the factor decreases. So long as this factor is greater than 1, T r..

Working rules for finding the greatest term: Step 1: In (a + b) n , the constants a and b must be positive. Step 2: Write T r+1 and T r and find the value of T r+1 /T r . Step 3: Simplify the inequality (T r+1 /T r ) greater than or equal to 1 and find the..

General Term for Positive Integral Index - For n N, we have . Let T r + 1 (0 r n) be the (r+1) t h term in the expansio..

Sometimes, a particular term satisfying certain conditions is required in the binomial expansion of the type (a + b) n . This can be done by expanding (a + b) n and then locating the required term. Generally this becomes a tedious task, specially when the index n is large. In such cases, we ..

For n N, we have . Let T r + 1 (0 r n) be the (r+1) t h term in the expansion. For 0 r n, we have T r + 1 = n C r a n - r b r..

n C 0 , n C 1 , ..... n C n are called binomial coefficients. n C 0 , n C 2 n C 4 , ..... are called even binomial coefficients. n C 1 , n C 3 , n C 5 .... are called odd binomial coefficients. In case of no ambiguity, the binomial coefficients n C 0 , n C 1 , ..... n C n ..

Alternative Proof of Binomial Theorem for Positive Integral Index (Combinatorial Method). We have, (a + b) n = (a + b) (a + b) ....... n times. The terms on the RHS are obtained by taking one letter from each factor and multiplying them together. Choosing 'a' from all the factors, ..