Question 2. Use induction to show that for any natural n ≥ 3, one can be written as a sum of n 1 distinct fractions of the form where k is a natural number.

Q: 10. Using the first principle of math induction, prove that, for n positive integer n³ + 2n is…

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Q: Prove using mathematical induction that n³ + 2n is divisible by 3 for all natural numbers n.

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Q: Which of the following rules is used to find the number of positive integers less than 1000 that are…

A: To Determine: which of the following rules is used to find the number of positive integers less than…

Q: (a) How many three-digit nos. can be formed from the digits 0.1.2.3.4.5 & 6 if each digit can be…

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Q: Show that for every integer n, n³ + 2 is not divisible by 4.

A: Here, we use rules of divisibility.

Q: Find a formula for 1/2+1/4+1/8+.+ 1/2h by examining the values of this expression for small values…

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Q: Find a reduction formula for ∫sinnxdx, n≥2.

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Q: Compute the sum 42 + 47 + 52 + . . . + 662

A: 42+47+52+⋯+662Given series is arithmetic sequence a1=42a2=47a3=52d=a2-a1=47-42=5

Q: How many seven-digit numbers can be formed with the first two digits odd and the last five digits…

A: The total number of digits from 0 to 9 is 10.

Q: Justify that every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q.

A: Let p be a prime number other than 2 and 3. If p is divided by 6, then by the quotient-remainder…

Q: Suppose ʼn is a positive whole number. Simplify [n+1)! — n! × n. On! O 1 On + 1

A: 'n' is a positive whole number. We have to solve  (n+1)!-n!×n Remember : n!=n.(n-1)........3.2.1

Q: Consider the set of all fractions equal to12. If you take any10 of those fractions, add their…

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Q: Find the sum of n terms of an AP whose nth term is given by an=5-6n

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Q: Give an example of (a) a composite number of the form 2º – 1, where p is an odd prime. (b) a…

A: b) Here we need to find an example of a composite number of the form  22n+1 where n is a non…

Q: Given *,=4, x,=7, x,=5, x,=9, x,=1I, x,=2, and x,=6 , evaluate the summation: 2*, i=1 A 34 B. 38 40…

A: Given,x1=4 , x2=7 , x3=5 , x4=9 , x5=11 , x6=2 , x7=6

Q: What is the sum 1+2+4+8+...+512?

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Q: Evaluate the sums

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Q: Prove: The sum of an odd number of odd numbers must be odd.

A: Let us prove this by induction, that if n is odd and a1,a2,. . .. .an are odd, then ∑i=1nai is odd.…

Q: Find the sum of 39+40+41+42+...+111+112

A: To find the sum of a series, we need to determine the number of terms in the series. Then we need to…

Q: Find 1 + 2 + 3 + 4 + ......... + 100, the sum of the first 100 natural numbers.

A: Given: The expression 1+2+3+...+100, To find: The sum of the expression.

Q: Ci) Use the method of Mathematical Induction to show that for all positive integer values n21,…

A: The principle of mathematical induction is a technique of proving a statement or formula which is…

Q: Prove by induction that it is possible to pay, without requiring change, any whole number of…

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Q: Find the sum of 1 + 3 + 9 + 27 + … to 12 terms.

A: =1+3+9+27+...+n.  in this a1=1,a2=3            r=a2/a1=3/1=3            s=(a(rn -1))/(r-1)…

Q: Show by induction: for all integers n > 5, 2" > n? +n .

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Q: For how many values of n is the sum 1 + 2 + · • + n not divisible by n if 1 ≤ n < 100?

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Q: 1+ 1.19+1.19 to the 2 power+1.19 to the 3rd power............+ 1.19 to the 17 power.

A: Solution-  Since it is in geometric progression, because ratio is common for all. It is in the form…

Q: 3. Find the Sum of the first 25 terms of: 32, 28, 24, 20,...

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Q: 2. In how many ways can three numbers be selected from the numbers 1, 2, ... , 300 such that their…

A: This problem is solved using the concept of permutation and combination. please find step by step…

Q: Show that if 5 divides n2 with remainder 4, 5 divides 3n3 with remainder 2.

A: Given that 5 divides n2 5 n25 n×n5 n5 n3 As it gives the remainder 4 on dividing by 5. Therefore it…

Q: Derive an expression for the sum of the first n positive even integers.

A: We Use the A.P series. Find The sum of the first n positive even integer.

Q: How many even 4-digit numbers can be formed from the digits 1, 2, 6, 7, and 9 with no repetitions…

A: There is no repetition allowed.. At unit place we have two choices 2 or 6 In 10th place we have…

Q: (a) Which primes can divide n² + 1 for some n? a 30000 40 AT & opinip ad seppo 90.4/3 (4) рро t' for…

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Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is P(0)…

A: Principles of mathematical induction : Algorithm: Obtained P(n) and understand it's meaning.  Prove…

Q: How do I prove: If n > 1 is a natural number then there is a prime number p such that p|n.

A: This is a theorem of real analysis in which we have to find that there is a p who divides n.