Introduction to Binomial, Exponential and Logarithmic series
A binomial is an algebraic expression of two terms. Now let us see the following binomial expansion and the number pattern we get adjacent to it.
From the above, we observe that the binomial coefficients form a number pattern which is in a triangular form. This pattern is known as Pascal’s triangle. [ In pascal’s triangle, the binomial coefficients appear as each entry is the sum of the two above it.]
For any natural number n
In the above binomial expansion, we observe,
(i) The (r+1)th term is denoted by T r+1 = ncr xn–rar
(ii) The degree of ‘x’ in each term decreases while that of ‘a’ increases such that the sum of the power in each term equal to n
(iii) nc0, nc1, nc2, … ncr, … ncn are binomial coefficients they are also written as c0, c1, c2 … , cn
(iv) From the relation ncr = ncn–r we see that the coefficients of term equidistant from the beginning and the end are equal.
Expand (2x+y)5 using binomial theorem.
Find the middle terms of expansion (3x+y)5
In the expansion of (3x+y)5 we have totally 6 terms. From this the middle terms are T3 and T4
Similarly we can find by putting r = 3 in Tr+1 to get T4 , then T4 =90x2y3
For any rational number other than positive integer
Find the approximate value of 3√1002 (correct to 3 decimal places) using Binomial series.