prove that the statement 1/5+1/5^2+1/5^3 +...+1/5^n=1/4(1-1/5^n) is true for all positive integers n, write your proof in the space below
Hint:
(1-x^n) = (1-x)(1+x+x²+x³+...+xn-1)
To prove the statement 1/5 + 1/5^2 + 1/5^3 + ... + 1/5^n = 1/4(1 - 1/5^n) holds true for all positive integers n, we will use mathematical induction.
Step 1: Base Case
Let's verify the statement for the base case, where n = 1.
1/5 = 1/4(1 - 1/5^1)
1/5 = 1/4(1 - 1/5)
1/5 = 1/4(5/5 - 1/5)
1/5 = 1/4(4/5)
1/5 = 1/5
The base case holds true since both sides of the equation are equal.
Step 2: Inductive Hypothesis
Assume that the statement holds true for an arbitrary positive integer k, i.e.,
1/5 + 1/5^2 + 1/5^3 + ... + 1/5^k = 1/4(1 - 1/5^k).
Step 3: Inductive Step
Now, we need to prove that if the statement is true for k, then it is also true for k+1.
1/5 + 1/5^2 + 1/5^3 + ... + 1/5^k + 1/5^(k+1) = 1/4(1 - 1/5^(k+1))
From the inductive hypothesis, we know that:
1/5 + 1/5^2 + 1/5^3 + ... + 1/5^k = 1/4(1 - 1/5^k).
Substituting this into the equation, we have:
(1/4)(1 - 1/5^k) + 1/5^(k+1) = 1/4(1 - 1/5^(k+1))
Simplifying the left side of the equation:
(1 - 1/5^k)/4 + 1/5^(k+1) = (5^k - 1)/4 * (5^k * 5)/(5^k * 5^k * 4)
Multiplying both sides by 5^(k+1) * 4:
(5^k - 1) * 5 + 4 = (5^k * 5)
Multiplying:
5^(k+1) - 5 + 4 = 5^(k+1)
Simplifying:
5^(k+1) - 1 = 5^(k+1)
Both sides of the equation are equal, which completes the inductive step.
Step 4: Conclusion
The proof by mathematical induction shows that the statement 1/5 + 1/5^2 + 1/5^3 + ... + 1/5^n = 1/4(1 - 1/5^n) holds true for all positive integers n.