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Question: Given in the expansion of \[{{\left( 1+x \right)}^{43}},\] the coefficient of \[{{\left( 2r+1 \right...

Given in the expansion of (1+x)43,{{\left( 1+x \right)}^{43}}, the coefficient of (2r+1)th{{\left( 2r+1 \right)}^{th}} and coefficient of (r+2)th{{\left( r+2 \right)}^{th}} terms are equal. Find the value of r.

Explanation

Solution

To solve the given question, we will first find out what binomial expansion of any term (a+b)m{{\left( a+b \right)}^{m}} is. Then we will try to write the general term of this expansion in terms of m, a, and b. After writing the general term, we will replace m with 43, a with 1 and b with x. Now, we will determine the coefficient of this general term. With the help of this, we will find the coefficient of (r+2)th{{\left( r+2 \right)}^{th}} term and (2r+1)th{{\left( 2r+1 \right)}^{th}} term. Then, we will form two cases. In case I, we will equate the coefficients of (r+2)th{{\left( r+2 \right)}^{th}} term and (2r+1)th{{\left( 2r+1 \right)}^{th}} term. From here, we will get a value of r. In case II, we will make use of the condition that if (i+1)th{{\left( i+1 \right)}^{th}} and (j+1)th{{\left( j+1 \right)}^{th}} term of (a+b)m{{\left( a+b \right)}^{m}} are equal, then i + j = m. From here, we will get another value of r.

Complete step-by-step answer:
To start with, let us first find out what binomial expansion is and what will be the binomial expansion of (a+b)m.{{\left( a+b \right)}^{m}}. The binomial expansion describes the algebraic expansion of powers of a binomial. The binomial expansion of (a+b)m{{\left( a+b \right)}^{m}} is given as shown below.
(a+b)m= mC0amb0+ mC1am−1b1+ mC2am−2b2+ mC3am−3b3+........+ mCm−1a1bm−1+ mCma0bm{{\left( a+b \right)}^{m}}={{\text{ }}^{m}}{{C}_{0}}{{a}^{m}}{{b}^{0}}+{{\text{ }}^{m}}{{C}_{1}}{{a}^{m-1}}{{b}^{1}}+{{\text{ }}^{m}}{{C}_{2}}{{a}^{m-2}}{{b}^{2}}+{{\text{ }}^{m}}{{C}_{3}}{{a}^{m-3}}{{b}^{3}}+........+{{\text{ }}^{m}}{{C}_{m-1}}{{a}^{1}}{{b}^{m-1}}+{{\text{ }}^{m}}{{C}_{m}}{{a}^{0}}{{b}^{m}}
Here, we can see that the first term is mC0amb0,^{m}{{C}_{0}}{{a}^{m}}{{b}^{0}}, the second term is mC1am−1b^{m}{{C}_{1}}{{a}^{m-1}}b and so on. Thus, we can say that the general term of this expansion would be
General term or pth term= mCp−1am−p+1bp−1\text{General term or }{{\text{p}}^{th}}\text{ term}={{\text{ }}^{m}}{{C}_{p-1}}{{a}^{m-p+1}}{{b}^{p-1}}
Thus, the general term of (1+x)43{{\left( 1+x \right)}^{43}} will be equal to
General term or pth term= 43Cp−1(1)43−p+1(x)p−1\text{General term or }{{\text{p}}^{th}}\text{ term}={{\text{ }}^{43}}{{C}_{p-1}}{{\left( 1 \right)}^{43-p+1}}{{\left( x \right)}^{p-1}}
⇒General term or pth term= 43Cp−1×1×xp−1\Rightarrow \text{General term or }{{\text{p}}^{th}}\text{ term}={{\text{ }}^{43}}{{C}_{p-1}}\times 1\times {{x}^{p-1}}
Now, the coefficient of this general term is 43Cp−1.^{43}{{C}_{p-1}}. Thus, we can say that the (2r+1)th{{\left( 2r+1 \right)}^{th}} term’s coefficient will be 43C2r+1−1= 43C2r.^{43}{{C}_{2r+1-1}}={{\text{ }}^{43}}{{C}_{2r}}.
Similarly, (r+1)th{{\left( r+1 \right)}^{th}} term’s coefficient will be 43Cr+2−1= 43Cr+1.^{43}{{C}_{r+2-1}}={{\text{ }}^{43}}{{C}_{r+1}}.
Now, we are given that these coefficients are equal. Thus, we have two cases.
Case 1: As both the coefficients are equal, we have,
43C2r= 43Cr+1^{43}{{C}_{2r}}={{\text{ }}^{43}}{{C}_{r+1}}
Thus, 2r = r + 1
Case 2: If mCa= mCb^{m}{{C}_{a}}={{\text{ }}^{m}}{{C}_{b}} then we can say that a + b = m. Thus, we will get,
⇒2r+r+1=43\Rightarrow 2r+r+1=43
⇒3r+1=43\Rightarrow 3r+1=43
⇒3r=42\Rightarrow 3r=42
⇒r=14\Rightarrow r=14
Thus, the values of r are 1 and 14.

Note: We can also write the binomial expansion of (a+b)m{{\left( a+b \right)}^{m}} as shown below.
(a+b)m= mC0a0bm+ mC1a1bm−1+ mC2a2bm−2+ ........+ mCmamb0{{\left( a+b \right)}^{m}}={{\text{ }}^{m}}{{C}_{0}}{{a}^{0}}{{b}^{m}}+{{\text{ }}^{m}}{{C}_{1}}{{a}^{1}}{{b}^{m-1}}+{{\text{ }}^{m}}{{C}_{2}}{{a}^{2}}{{b}^{m-2}}+\text{ }........+{{\text{ }}^{m}}{{C}_{m}}{{a}^{m}}{{b}^{0}}
It does not depend on how we write the expansion of (a+b)m,{{\left( a+b \right)}^{m}}, the answer will be the same because the coefficients of the expansion of (a+b)m{{\left( a+b \right)}^{m}} have the property mCp= mCm−p.^{m}{{C}_{p}}={{\text{ }}^{m}}{{C}_{m-p}}.