Binomial Theorem Application for Positive Integral Index

Ask a Question, Get an Answer!

Type your question here

Hundreds of tutors are online and ready to help you right now!

Theorem

Using Binomial theorem, prove that:

Proof:

When x = 1,

When x = -1

\ Hence (i) is proved.

\ Hence (ii) is proved.

\ Hence (iii) is proved.

Note 1:

The L.H.S of (ii) and (iii) will have only finite number of terms, because ⁿC_r = 0 in case r>n.

Note 2:

ⁿC₀, ⁿC₁, ..... ⁿC_n are called binomial coefficients. ⁿC₀, ⁿC₂ ⁿC₄, ..... are called even binomial coefficients. ⁿC₁, ⁿC₃, ⁿC₅ .... are called odd binomial coefficients.

Note 3:

In case of no ambiguity, the binomial coefficients ⁿC₀, ⁿC₁, ..... ⁿC_n are written as C₀, C₁, ..... C_n.

Note 4:

The last term in L.H.S of (ii) and (iii)will depend on the fact as to whether n is even or odd.

Working rules for solving problems:

Rule 1:

If n N, then

Rule 2:

If n N, then

Rule 3:

If n N, then