Question

What is the sum of the coefficients of the expansion of (2x -1)^20?​

Answer 1

Answer:

Step-by-step explanation:

The index 20 in (1 + x) 20 in (1 +x) 20 is even.  Middle term = T 20 + 2 2 T20+22  = T10 + 1 = 20C10 (1)10 x10  = 20C10 x10  Now coefficient of middle term in (1 +x) 19  The index 19 in (1 +x) 19 is odd.  So, middle term as T 19 + 1 2 T19+12 and the next term i.e., T10 and T11 T10 = = T9 + 1 = 19C9 × 110 × x9 = 19C9 x9  And T11 = T10 + 1 = 19C10 × x10 = 19C10 x10 Now, sum of coefficient of middle terms in (1 + x) 19 = 19C9 + 19C10 = 20C10 = coefficient of middle term in (1 + x) 20  = Hence Proved. ← Prev QuestionRead more on Sarthaks.com - https://www.sarthaks.com/1029551/show-that-the-coefficient-of-middle-term-in-the-expansion-of-equal-the-sum-the-coefficients

Answer 2

The sum of the coefficients of the expansion of (2x -1)^20 is 1.

Given:

(2x -1)^20

To find:

The sum of the coefficients of the expansion of (2x -1)^20.

Solution:

Binomial expansion is the expansion of an expression which is raised to some power using the formula:

(a+b)ⁿ= ⁿC₀(a)ⁿ(b)⁰+ ⁿC₁(a)ⁿ⁻¹(b)¹+ⁿC₂(a)ⁿ⁻²(b)²+ⁿC₃(a)ⁿ⁻³(b)³+..................+ ⁿCₙ(a)⁰(b)ⁿ

As we know the Binomial expression of  (2x -1)²⁰ is

(2x -1)²⁰ = ²⁰C₀(2x)²⁰ + ²⁰C₁(-1)¹(2x)⁹⁸ + ²⁰C₂(-1)²(2x)⁹⁷+ ......+²⁰C₂₀(-1)²⁰

Now to find the sum of all coefficients in the expression we put x =1. Because that will add all the coefficients in the expression , so the sum will be  (2x -1)²⁰ = ²⁰C₀(2x)²⁰ + ²⁰C₁(-1)¹(2x)¹⁹ + ²⁰C₂(-1)²(2x)¹⁸+ ......+²⁰C₂₀(-1)²⁰

Thus, sum of coefficients  = [(2×1)-1]²⁰ = 1²⁰= 1

Therefore, the sum of the coefficients of the expansion of (2x -1)^20 is 1.

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