Prove by mathematical induction that for all nonnegative integers n that                                                                                                                                     1 + 1/2 + 1/4 + 1/8 +· · ·+ 1/2n = 2 − 1/2n

Q: 16- Prove that for every positive integer n and all real number r except 1: -1 r -1 i=0 17- Show…

A: We need to prove the following given condition in question 16 and 17

Q: Prove by induction that 7 divides 3^(5n+3) + 5^n for every non-negative integer n.

A: We can prove it by using mathematical induction.

Q: Use mathematical induction to prove that (3n + 7n − 2) is divisible by 4 for all integers n ≥ 1.

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Q: If n2 4 is an integer,prove by induction that n<nl.

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Q: Prove that by mathematically Induction 2n < (n + 1)!, for all integers n ≥ 2.

A: We have to prove that by principle of mathematical Induction2n &lt; (n + 1)!, for all integers n ≥…

Q: Prove by induction that n3 -n is divisible by 6 for all positive integers

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Q: Use induction to prove that for all n>0 in natural numbers , 1/1+1/4+1/9+...+1/n^2 is less than or…

A:

Q: Prove that by mathematically Induction 2n < (n + 1)!, for all integers n ≥ 2.

A: Let s(n):2n&lt;(n+1)!,n≥2...(1) Now, Substitute n=1 in (1), so, 2(2)≤(2+1)!⇒4≤3!⇒4&lt;6 Which is…

Q: Using proof by induction

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Q: (2) Prove by mathematical induction that for all integers n > 0, the number 42n+1 + (21 · 33"+1) is…

A:

Q: Prove by induction that for all integers n > 1, n n(n+ 1)(2n + 1) 6. i=1

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Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

Q: Prove that 2n &gt; n for all positive integers n.

A:

Q: Prove: for every positive integer n, 2"> n

A: As per the guidelines, since there are more than one question only one question will be answered.…

Q: Prove by mathematical induction that 3n+2n+3 is divisible by 4 for all positive integers

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Q: -ん 2 kodd

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Q: Prove or disprove: 2n + 1 is prime for all nonnegative integers n.

A: Prove or disprove: 2n + 1 is prime for all nonnegative integers n.

Q: Prove using Mathematical induction that for any positive integer 2 + 4 +6+... +2n=n(n+ 1)

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: Let n be a positive integer. Prove that (:) n +2. +...+n. n2"-1.

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Q: snip

A: To Prove n! ≥ 2n for all intergers n≥4We may prove this by Induction Method.

Q: Prove that 6 divides n3-n for all non-negative integers n.

A: See the detailed solution below.

Q: Prove by mathematical induction that for all positive integers n, 1 + 3 + 5 + … + (2n – 1) = n^2.

A: We will prove the given statement using mathematical induction method.

Q: Use mathematical induction to prove that 12 + 32 + 52 + .…+ (2n + 1)² = (n+1)(2n+1)(2n+3) where is a…

A: We need to prove the given result by mathematical induction. The solution is as follows.

Q: Q1 Use mathematical induction to prove that (-5)"+1 – 1 3 + 3 * (-5) + 3 * (-5)² + 3 * (-5)³ + .… +…

A: Given equation is, 3+3(-5)+3(-5)2+3(-5)3+......+3(-5)n=(-5)n+1-1-2

Q: Let n be a positive integer. Prove that :)- (:)- +2. +...+n. = n2"-1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: Prove that by mathematically Induction a. 2n < (n+ 1)!, for all integers n2 2.

A:

Q: Prove by mathematical induction that for every positive integer n, 1 1 1 1 П 3i 3і Зі + 1 (Зп + 1)!…

A: Since you have posted multiple questions and we can answer one question at a time. So, answered…

Q: Prove that by mathematically Induction 2" < (n + 1)!, for all integers n2 2.

A:

Q: Prove by induction that (2n+1)3 is an odd number for all positive integers n.

A: First principle mathematically induction : Suppose is a mathematical statement .Here n is a natural…

Q: Use mathematical induction to prove that 12 + 22 + 32 + ...... + n2 = n(n + 1)(2n + 1)/6 for all…

A: 12+22+32+.......+n2=nn+12n+16 Step 1: For n=1, Left hand side=12=1 &amp; Right hand…

Q: 5) Prove by induction on n that, for all positive integers n, 21|(4n+1+52n-1).

A:

Q: 3. Prove by either Well Ordering Principle or induction that for all nonnegative integers,n: + 1) i3…

A: We prove by induction .Case n=1LHS=∑i3i=01=03+13=1RHS=11+122=222=1Induction hypothesis : Assume…

Q: Prove by mathematical induction that for all positive integers n 1: a 12 x 2 + 22 × 3 + 32 × 4 + ·…

A: Let P(n) be the given statement. Then P(n):12x2 +22x3 + ... +n2 (n+1)= (1/12)n(n+1)(n+2)(3n+1) Thus…

Q: 2. Prove by induction that for all integers n > 0 5n+1 - 1 1+5+52 + ... + 5" 4

A: prove of the above by induction is given below...

Q: Prove by induction that for positive integers n:  4|(5^(n+1)+ 3^(2n-1)).

A:

Q: 12 + 22 + ...+ n² = n(n+1)(2n + 1) 6

A:

Q: Use induction to prove that 16 divides 5n+2 + 13"+3 – 14 - for all n > 0.

A:

Q: Use mathematical induction to prove the formula for all integers n2 1. 2+ 11 + 20 + 29 + … + (9n –…

A: Solution is in step 2

Q: Use mathematical induction to prove that for all n ≥ 2, if the product of n integers is odd, then…

A:

Q: prove that there is no positive integer n such that n^2+n# = 100

A: n^2+n^3 = 100 We can evaluate this till some positive integers

Q: Prove 8^ +l is composic for any poSitive integer n.

A:

Q: Prove by mathematical induction that for any integer n>0, 1 3 X4 X5 + 2 4x5x6 + + n = (n + 2)(n+3)(n…

A:

Q: Prove that by mathematically Induction (Marks = 05) 2n < (n + 1)!, for all integers n ≥ 2.

A: Answer

Q: 1. Prove by mathematical induction, for any positive integer n, n³ + 2n is divisible by 3.

A: as per company guidelines we are supposed to answer only one question with maximum three subparts.…

Q: Prove that there is an infinity of natural numbers n such that 2n +1 is divisible by n3

A: Solution of the problem as follows

Q: Prove by induction that for positive integers n: 1! x3! x5! x...(2n (2n-1):2 (n!)"

A: For n=1 Left hand side = 1!=1 Right hand side  =n!n=1!1=1 now 1≥1 The result is true for n=1 Assume…

Q: Prove by mathematical induction that 2n =0

A:

Q: Prove by induction that for any integer n: n(n+1) Σ; 2 j=1

A:

Q: Prove that n! > n2 for every integer n ≥ 4.

A: Use mathematical induction, to prove the statement n!&gt;n2 is true for all integer n≥4. Substitute…

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.