Expansion of (a+b)^n

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Method of writing expansion for (a+b)ⁿ

The first term in the expansion of (a + b)ⁿ is ⁿC₀aⁿb⁰. For the second term, the coefficient is taken as ⁿC₁, the power of 'a' is decreased by one and the power of 'b' is increased by one. So, the second term is ⁿC₁a^n-1b¹. For the third term, the coefficient is taken as ⁿC₂, the power of 'a' is again decreased by one and the power of 'b' is increased by one. This process goes on, till we get the last term as ⁿC_na⁰bⁿ.

Some Observations

For n N, in the expansion of (a + b)ⁿ, we observe that:

• the number of terms is n+1

• the exponent of 'a' decreases from n to 0

• the exponent of 'b' increases from 0 to n

• the sum of exponents of 'a' and 'b' in any term is n.

• the coefficient of any term is ⁿC_k where k is the exponent of 'b'.

• ⁿC₀, ⁿC₁, ⁿC₂, ..... ⁿC_n are called the binomial coefficients.

Since, ⁿC_r = ⁿC_n-r, we have ⁿC₀ = ⁿC_n, ⁿC₁ = ⁿC_n-1, ⁿC₂ = ⁿC_n-2, ....

The binomial coefficients in (a + b)ⁿ, which are equidistant from beginning and end are equal.

The binomial coefficients in the expansion of (a + b)ⁿ can be easily evaluated by using the Pascal's triangle given below:

In the Pascal's triangle, each row starts and ends with 1 and each coefficient in a row is equal to the sum of the two coefficients one just before it and other just after it in the preceding row.