Binomial Theorem, Sequences and Series: Introduction

Introduction

Binomial theorem facilitates the algebraic expansion of the binomial (a + b) for a positive integral exponent n. Binomial theorem is used in all branches of Mathematics and also in other Sciences. Using the theorem, for example one can easily find the coefficient of x²⁰ in the expansion of (2x−7)²³. If one wants to know the maturity amount after 10 years on a sum of money deposited in a nationalised bank at the rate of 8% compound interest per year or to know the size of population of our country after 15 years if the annual growth rate and present population size are known, Binomial theorem helps us in finding the above quantities. The coefficients that appear in the binomial expansion of (a + b)ⁿ, n ∈ N, are called binomial coefficients. Binomial theorem plays a vital role in determining the probabilities of events when the random experiment involves finite sample space and each outcome is either success or failure. In this chapter we learn binomial theorem and some of its applications.

Greek Mathematician Euclid mentioned the special case of binomial theorem for exponent 2. Binomial theorem for exponent 3 was known by 6^th century in India. In 1544, Michael Stifel (German Mathematician) introduced the term binomial coefficient and expressed (1 + x)ⁿ in terms of (1 + x)ⁿ⁻¹.

Over the period of thousand years, legends have developed mathematical problems involving sequences and series. One of the famous legends about series concerns the invention of chess, where the cells of chess board were related to 1, 2, 4, 8, . . . (imagine the number related to 64^th cell). There are many applications of arithmetic and geometric progressions to real life situations.

In the earlier classes we have learnt about sequences, series. Roughly speaking a sequence is an arrangement of objects in some order and a series is the sum of the terms of a sequence of numbers. The concept of infinite series helps us to compute many values, like sin 9/44 π, log 43 and e²⁰ to a desired level of approximation. Sequences are important in differential equations and analysis. We learn more about sequences and series.