consecutive positive numbers

They are integers that differ from each other by one, e.g., 8, 9, 10 are consecutive integers, also 180, 179, 178, 177 are consecutive intege..

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..

General Term for Positive Integral Index is: For 0 r n, we have T r + 1 = n C r a n - r b r..

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, ..