Question

Prove the following claim by using induction on n: Let 0 < α be a constant....

Prove the following claim by using induction on n: Let 0 < α be a constant. Then α^0 + α^1 + α^2 ... α^n = (1-α^(n+1)) / (1 - α) You can assume α DNE 1.

Next, assume in addition that α < 1, and that n = ∞. Could you simplify the equation in this case?

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Prove the following using induction or Induction Principle: Given Tautologies α(alpha), β(beta). Prove that (α and...
Prove the following using induction or Induction Principle: Given Tautologies α(alpha), β(beta). Prove that (α and β) is also a tautology. Provide clear proof.
True-False-Prove-Salvage. Let a ∈ N (so a is a constant). Claim: a|((a +1)^n −1) for n...
True-False-Prove-Salvage. Let a ∈ N (so a is a constant). Claim: a|((a +1)^n −1) for n ≥ 0.
Prove the following statement by mathematical induction. For every integer n ≥ 0, 2n <(n +...
Prove the following statement by mathematical induction. For every integer n ≥ 0, 2n <(n + 2)! Proof (by mathematical induction): Let P(n) be the inequality 2n < (n + 2)!. We will show that P(n) is true for every integer n ≥ 0. Show that P(0) is true: Before simplifying, the left-hand side of P(0) is _______ and the right-hand side is ______ . The fact that the statement is true can be deduced from that fact that 20...
) Let α be a fixed positive real number, α > 0. For a sequence {xn},...
) Let α be a fixed positive real number, α > 0. For a sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that {xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all n). (b) Prove that {xn} is bounded from below. (Hint: use proof by induction to show xn > √ α for all...
Problem 3. Let n ∈ N. Prove, using induction, that Σi^2= Σ(n + 1 − i)(2i...
Problem 3. Let n ∈ N. Prove, using induction, that Σi^2= Σ(n + 1 − i)(2i − 1). Note: Start by expanding the righthand side, then look at the following pyramid (see link) from
Automata Please prove the following by induction. Let S(n) be the sum of squares from 1...
Automata Please prove the following by induction. Let S(n) be the sum of squares from 1 to n, i.e., S(n)=1^2 + 2^2 + 3^2 + ... + n^2 Then S(n) = n(n+1)(2n+1)/6 = (2n^3+3n^2+n)/6
Using induction, prove the following: i.) If a > -1 and n is a natural number,...
Using induction, prove the following: i.) If a > -1 and n is a natural number, then (1 + a)^n >= 1 + na ii.) If a and b are natural numbers, then a + b and ab are also natural
Structural Induction on WFF For a formula α ∈ WFF we let `(α) denote the number...
Structural Induction on WFF For a formula α ∈ WFF we let `(α) denote the number of symbols in α that are left brackets ‘(’, let v(α) the number of variable symbols, and c(α) the number of symbols that are the corner symbol ‘¬’. For example in ((p1 → p2) ∧ ((¬p1) → p2)) we have l(α) = 4, v(α) = 4 and c(α) = 1. Prove by induction that he following property holds for all well formed formulas: •...
a) Let R be an equivalence relation defined on some set A. Prove using induction that...
a) Let R be an equivalence relation defined on some set A. Prove using induction that R^n is also an equivalence relation. Note: In order to prove transitivity, you may use the fact that R is transitive if and only if R^n⊆R for ever positive integer ​n b) Prove or disprove that a partial order cannot have a cycle.
Structural Induction on WFF For a formula α ∈ WFF we let l(α) denote the number...
Structural Induction on WFF For a formula α ∈ WFF we let l(α) denote the number of symbols in α that are left brackets ‘(’, let v(α) the number of variable symbols, and c(α) the number of symbols that are the corner symbol ‘¬’. For example in ((p1 → p2) ∧ ((¬p1) → p2)) we have l(α) = 4, v(α) = 4 and c(α) = 1. Prove by induction that he following property holds for all well formed formulas: •...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT