positive natural numbers

For any rational number n, We accept this expansion without proof. The restriction on x is not required when n is a natural number. Now, we shall see that when n is a natural number, then the above expansion coincides with that as given earlier. Let..

The countable numbers are called Natural Numbers. Therefore 1 is the smallest or least of all natural numbers. The highest natural number cannot be said because Natural numbers are infinite. The se..

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 ..