Algebra-2

To find the middle term in the expansion of using the binomial expansion and Pascal's Triangle, we can follow these clear steps:

Step 1: Identify the Expression

We are expanding:

 

Step 2: Total Number of Terms

The total number of terms in the expansion is (n + 1):

n + 1 = 6 + 1 = 7

Since there are 7 terms, the middle term corresponds to the 4th term.

 

Step 3: Determine the Fourth Term

The binomial expansion can be expressed as:

Where:

a = x

b = 2

n = 6

The 4th term (which corresponds to in the binomial expansion) can be expressed as:

Fourth Term = 

This can be rewritten as:

Fourth Term =

 

Step 4: Constructing Pascal's Triangle

To find the coefficient , we need the 7th row of Pascal's Triangle (row indices start from 0):

Row 0:           1

Row 1:         1   1

Row 2:       1   2   1

Row 3:     1   3   3   1

Row 4:   1   4   6   4   1

Row 5: 1   5  10  10   5   1

Row 6: 1   6  15  20  15   6   1

 

Step 5: Finding the Coefficient

In the 7th row (row 6), the coefficients are:

1, 6, 15, 20, 15, 6, 1

 

The coefficient for the 4th term (which is

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