Find the 40th term of the series 1000 + 940 + 880 + 820 +... a) -1340 b) -2400 c) 400

Q: Prove using mathematical induction that (n + 1)! > 2"-1, for any integer n > 1.

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Q: Define the sequence {b, } as follows: bo = 4 %3D b, = bn-1 +2n + 3 for n 2 1 %3D Use induction to…

A: The given sequences bn as follows:                                                         b0=4…

Q: Establish formula for the sum of First n odd positive integers and Prove it by using Mathematical…

A: Let 2n-1 represent any odd number , where n is a natural number.We need to find the sum of the…

Q: Show that in a sequence of m integers there exists one or more consecutive terms with a sum…

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Q: 62. Prove part (b) of Theorem 4.4 in the case when n is odd: If n(n+1) n is a positive odd integer,…

A: ∑k=1nk=n(n+1)2 , n is odd integer

Q: Use mathematical induction to show that any postage of at least 6 cents can be obtained using 2e and…

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Q: Prove with induction the sum of first n Fibonacci numbers with even indices, i.e., f2 +f4 +···+f2n…

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Q: Find 41 term of series 12,18,24..

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Q: A sequence {a,} is defined recursively by aj = 1, az = 4, a3 = 9 and an = an–1 – an–2+ an-3 + 2(2n –…

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Q: 2. Prove that the number of irreducible fractions not greater than 1 and with denominator equal to n…

A: For the solution of the problem follow the next step.

Q: 9. Use Cantor's diagonal argument to show that the set of all infinite sequences of Os and 1s (that…

A: please see the next step for solution

Q: Prove the formula for the sum of the first positive integers:

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Q: Use ibn al-Haytham's procedure to derive the formula for the sum of the fifth powers of the…

A: We have to derive the formula 15+25+......+n5=n66+n52+5n412−n212 by using ibn al-Haytham's method…

Q: 4. There is a stack of logs in the backyard. 15 logs are in the first layer; 14, in the second; 13,…

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Q: Prove that the sum of the 1st natural n numbers = n*(n+1)/2

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Q: 3 3 +2 +3 +...n ninti)73 for all n21

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Q: 3. Let n be an integer, show that if n is even, then n is also even using indirect proof.

A: Given : Let n be an integer.   To show that if n2 is even, then n is also even.

Q: What is the sum of the first 19 terms of the series 2 - 5 + 12.5 - 31.25 + ..

A: Given series is2-5+12.5-31.25+.... The sum of the first 19 terms of the series is determined as…

Q: Establish formula for the sum of first n odd positive integers and prove it by using mathematical…

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Q: Prove by induction that the sum of the first n integers starting at 1 (1,2,3,...n) is n(n+1)/2.

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Q: A sequence ao, a1, a2, .. .satisfies an = n an-1+(-1)" for n > 1. If a5 = 44, then az %3D

A: Since, a sequence a0,a1,a2,..... satisfies an=n.an-1+(-1)n-(i) for n≥1.Ifa5=44. Our Aim is to find…

Q: 1. How many positive integers of 5-digit-number can be formed from the prime numbers starting at 1…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Define the sequence {b,} as follows: bo = 1 bn = bn-i +n for n > 1 %3D n(n + 1) Use induction to…

A: Given- Define the sequence bn as follows: b0 = 1bn = bn-1 + n for n ≥ 1 To find- Use Induction to…

Q: 5-a) How many prime triples of the form (p, 2p + 1, 4p + 1) are there? Describe them all, if…

A: a) The objective is to state the number of prime triplets of the form p,2p+1,4p+1. b) To verify the…

Q: 11**) (Solution to be discussed in tutorial only) For a polygon to be conver means that all of its…

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Q: Using the idea for the construction of the Fibonacci numbers, complete the list of the first 15…

A: Using the idea for the construction of the Fibonacci numbers, complete the list of the first 15…

Q: Find the 70th term of the following arithmeti c sequence. 4, 11, 18, 25

A: First term of the A.P. is 4 Common difference, means the difference between two consecutive terms is…

Q: if the 8th term in an arithmatic series is 39 and the 20th term -33 what is the common difference

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Q: or an integer n>=3, an n-gon is an n-sided polygon. (example, a 3-gon is a triangle, 4-gon is a…

A: To prove: the sum of the interior anles of every polygon with n sides is n-2180° Now we shall take a…

Q: In each case, find the first 5 terms of the Sum and comgute S

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Q: 3. Consider the pattern manifested by the tiles below. az az a4 as (a) What is the nth-term formula…

A: To find first five terms of the sequence,we substitute n=1,2,3,4,5 in the expression of Tn. After…

Q: Number theory

A: Consider the provided question, We have to prove that,

Q: Prove that the nth Fibonacci number fn≤αn−1, where α is the golden ratio.

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Q: 3. Prove that if 2 |un, then 4 | (u2 4. For the Fibonacci sequence, establish the following: 19ª is…

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Q: 3. Use principle of mathematical induction to prove that (n)(n + 1) 1+ 2+3+ ·..+n = for n > 1.

A: Solution

Q: Use any method to deter (a) 13 The series 8.

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Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Prove using Mathematic Induction: 5" – 1 is divisible by 4 for all natural numbers n.

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Q: 2: Assume that the patterns shown by the square tiles in the following figure continues. What is the…

A: Given that:   There are 5 pattern shown in the given figure, and find the total number of tiles in…

Q: 1. Define a sequence a1, a2, a3, ... by aj = = 1, an = 3an-1 + 4 for n> 2. Give a proof by induction…

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Q: For any n e N, 3 divides n – n. -

A: Prove using Mathematical of induction

Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

Q: positive

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Q: Suppose that ∞     n = 1an = 1, that ∞     n = 1bn = −1, that a1 = 2, and b1 = −3. Find the sum of…

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Q: 3. Use mathematical induction to prove that the sum of the cube of the first n positive integers…

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Q: Let SC R. Give necessary and sufficient conditions on S so that a sequence (r) of real numbers is…

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Q: Suppose that a sequence is defined by a1 = 1, an+1 =(a, + 8). To show that a, is monotonic using…

A: Notice a2 = 9/5 =1.8 > 1  So, we assume an will be increasing sequence.

Q: 1. How many terms of the arithmetic series 2,8,14,20,... are required to give a sum of 660?

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Q: How many terms are there in a series -64 -32 -16 -....-1÷32

A: Topic:- sequence and series