Binomial Coefficients with G.P. or A.P. Multiplied

The positive integers that are present as coefficients in the binomial expansion are termed binomial coefficients. It was introduced in the year 1826. The Indian mathematician Bhaskaracharya was the person who gave the description of the coefficients in the binomial expansion. The binomial coefficients can be computed using the following formulae:

1) Recursive formula: ⁿC_k = ^n-1C_k-1 + ^n-1C_k for all integers n.

2) Multiplicative formula:

\(\begin{array}{l}{\binom {n}{k}}={\frac {n^{\underline {k}}}{k!}}={\frac {n(n-1)(n-2)\cdots (n-(k-1))}{k(k-1)(k-2)\cdots 1}}=\prod _{i=1}^{k}{\frac {n+1-i}{i}}\end{array} \)

3) Factorial formula:

\(\begin{array}{l}{\binom {n}{k}}={\frac {n!}{k!\,(n-k)!}}\quad {\text{for }}\ 0\leq k\leq n\end{array} \)

4) Pascal’s triangle consists of the value of binomial coefficients.

Binomial Coefficients with G.P. or A.P. Multiplied

Basic idea:

Consider (1 + x)ⁿ = ⁿC₀ + ⁿC₁ x + ⁿC₂ x² + ⁿC₃ x³ + …….

Where x¹, x², x³ ….. are in G.P. multiplied with ⁿC₀, ⁿC₁, ….

Start from ⁿC₀.
x is a variable multiplied by ⁿC_1.

Example 1: ⁿC₀ + ⁿC₁ 3 + ⁿC₂3² + ⁿC₃ 3³ + …..

Solution:

By comparing the given problem to (1 + x)ⁿ = ⁿC₀ + ⁿC₁ x + ⁿC₂ x² + ⁿC₃ x³ + ……., the answer is (1 + 3)ⁿ.

Example 2: ⁿC₀ + ⁿC₁ 2¹ + ⁿC₂ 2² + ……

Solution:

It is a G.P. with a common ratio 2 = x

ⁿC₀ + ⁿC₁ 2¹ + ⁿC₂ 2² + ……

= (1 + 2)ⁿ

= 3ⁿ

Example 3: ⁿC₀ + ⁿC₁ 4² + ⁿC₂ 4⁴ + ……

Solution:

It is a G.P. with a common ratio 4² = x

ⁿC₀ + ⁿC₁ 4² + ⁿC₂ 4⁴ + ……

(1 + 4²)ⁿ = 17ⁿ

The case when AP is multiplied by the binomial coefficient.

ⁿC₀ a + ⁿC₁ (a + d) + ⁿC₂ (a + 2d) + ……… + ⁿC_n (a + nd)

= {[a (a + nd)] / 2} {ⁿC₀ + ⁿC₁ + ………. + ⁿC_n}

= {[a (a + nd)] / 2} (2ⁿ)

= a + (a + nd) 2^n-1

Proof:

S = ⁿC₀ a + ⁿC₁ (a + d) + ⁿC₂ (a + 2d) + ….. + ⁿC_n (a + nd)

S = ⁿC_n (a + nd) + ⁿC_n-1 [(a + (n – 1) d] + ……. + ⁿC₀ a

On adding the above two expansions,

2S = ⁿC₀ [a + (a + nd)] + ⁿC₁ [a + (a + nd)] + ⁿC_n-2 [a + (a + nd)] + …… + ⁿC_n [a + (a + nd)]

= [a + (a + nd)] [ⁿC₀ + ⁿC₁ + ⁿC₂ + ……. + ⁿC_n]

S = {[a (a + nd)] / 2} (2ⁿ)

Example 1: ⁿC₀ + 5 . ⁿC₁ + 7 . ⁿC₂ + …….. + (2n + 3) . ⁿC_n = ?

Solution:

{[3 + (2n + 3)] / 2} {2^n-1}

= (2n + 6) 2^n-1

= (n + 3) 2ⁿ

Example 2: Coefficient of x^r [0 ≤ r ≤ (n – 1)] in the expansion of

\(\begin{array}{l}{{(x+3)}^{n-1}}+{{(x+3)}^{n-2}}(x+2)+{{(x+3)}^{n-3}}{{(x+2)}^{2}}+…+{{(x+2)}^{n-1}}\end{array} \)

Solution:

\(\begin{array}{l}(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^{2}+….+(x+2)^{n-1}\end{array} \)

\(\begin{array}{l}=\frac{(x+3)^{n}-(x+2)^{n}}{(x+3)-(x+2)} = (x+3)^{n}-(x+2)^{n}\end{array} \)

\(\begin{array}{l}(\because \frac{x^{n}-a^{n}}{x-a})=x^{n-1}+x^{n-2}a^{1}+x^{n-3}a^{2}+….+a^{n-1}\end{array} \)

Coefficient of x^r in the given expression = Coefficient of x^r in [(x + 3)ⁿ – (x + 2)ⁿ]

\(\begin{array}{l}= ^{n}C_{r}3^{n-r}\; -\; ^{n}C_{r}2^{n-r}\end{array} \)

\(\begin{array}{l}^{n}C_{r}(3^{n-r} – 2^{n-r})\end{array} \)

Example 3: The sum of the last eight coefficients in the expansion of (1 + x)¹⁵is

Solution:

\(\begin{array}{l}^{15}{{C}_{0}}+{{\,}^{15}}{{C}_{1}}+…+{{\,}^{15}}{{C}_{15}}={{2}^{15}}\\ \Rightarrow \,\,\,2{{(}^{15}}{{C}_{8}}+{{\,}^{15}}{{C}_{9}}+…+{{\,}^{15}}{{C}_{15}})={{2}^{15}} (\because \,{{\,}^{n}}{{C}_{r}}={{\,}^{n}}{{C}_{n-r}})\\ ^{15}{{C}_{8}}+{{\,}^{15}}{{C}_{9}}+…+{{\,}^{15}}{{C}_{15}}\\ ={{2}^{14}}\end{array} \)

Example 4:

\(\begin{array}{l}\text{If}\ {{a}_{k}}=\frac{1}{k(k+1)}, \text{for}\ k=1,\,2,\,3,\,4,…..,\,n,\ \text{then}\ {{\left( \sum\limits_{k=1}^{n}{{{a}_{k}}} \right)}^{2}}=\end{array} \)

Solution:

\(\begin{array}{l}\sum\limits_{k=1}^{n}{{{a}_{k}}}=\sum\limits_{k=1}^{n}{\frac{1}{k\,(k+1)}}\\ =\left( 1-\frac{1}{2} \right)+\left( \frac{1}{2}-\frac{1}{3} \right)+…+\left( \frac{1}{n}-\frac{1}{n+1} \right)\\ = 1-\frac{1}{n+1}=\frac{n}{n+1}{{\left( \sum\limits_{k=1}^{n}{{{a}_{k}}} \right)}^{2}}\\ ={{\left( \frac{n}{n+1} \right)}^{2}}\end{array} \)

Binomial Coefficients with G.P. or A.P. Multiplied – Video Lesson

JEE Maths

Frequently Asked Questions

Q1

What do you mean by binomial coefficients?

Binomial coefficients are the positive integers which are the coefficients in the binomial theorem.

Q2

Give the binomial theorem formula.

The binomial theorem formula is given by (x+y)ⁿ = Σ_r=0ⁿ ⁿC_r x^{n – r} y^r. n ∈ N, x,y,∈ R.

Q3

Give two applications of the binomial theorem.

The binomial theorem is used to find the expansions for the algebraic identities. It is also used in probability for binomial expansion.

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