Question6. Find the positive value of $x$ that solves the following equation: $x^{\infty}=\sum_{k=0}^{30}\binom{30}{k} 20^{30-k}$
ANS: $\qquad$

Studdy Solution

STEP 1

1. The equation involves an infinite power, $x^{\infty}$ , which implies that $x$ must be 1 for the expression to be finite and non-zero.
2. The right-hand side of the equation is a finite sum involving binomial coefficients and powers of 20.

STEP 2

1. Analyze the left-hand side of the equation.
2. Analyze the right-hand side of the equation.
3. Determine the value of $x$ .

STEP 3

Analyze the left-hand side of the equation $x^{\infty}$ .
If $x^{\infty}$ is finite and non-zero, then $x$ must be equal to 1. This is because any number greater than 1 raised to the power of infinity becomes infinite, and any number less than 1 raised to the power of infinity becomes zero.

STEP 4

Analyze the right-hand side of the equation:
The right-hand side is a finite sum given by:
$\sum_{k=0}^{30} \binom{30}{k} 20^{30-k}$
This expression represents the expansion of $(1 + 20)^{30}$ using the binomial theorem, which states:
$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
Here, $a = 1$ , $b = 20$ , and $n = 30$ , so:
$(1 + 20)^{30} = 21^{30}$

STEP 5

Determine the value of $x$ :
Since $x^{\infty} = 21^{30}$ and $x^{\infty}$ is finite and non-zero, $x$ must be equal to 1.
Therefore, the positive value of $x$ that satisfies the equation is:
$\boxed{1}$

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