If m times the nth term of an AP is equal to n times its nth term, show that the (m + n) th term of AP is Zero

Q: If x is any positive integer , then the sum of 6x and 11x is always divisible by what?

A: x is a positive integer. Find sum of 6x and 11x is divisible by what?

Q: We need to find all the whole numbers that have remainder 1 upon division by 4 and write our answer…

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Q: Justify whether it is true to say that the given is the nth term of an AP  1+n+n2

A: We have to say that whether the given nth term is in A.P. where, 1+n+n2 We have given nth term of…

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

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Q: Is the statement true or false? If an odd number is less than 15, then it is prime. False Give a…

A: We have to check

Q: For positive integers m > n , show that -2 + n- 1 -3 + ...+ n + n-1 n - 1 + in

A: The right-side of the given equation can be expressed as…

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

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Q: If 6th term of a G.P. is 9375 and its 4th term is 375.Find its 9th term.

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Q: Find the smallest positive integer that leaves a remainder of 1 when divided by 2, a remainder of 2…

A: Let N be such a number N=10a+9:  N+1 = 10a+10 N=9b+8: N+1 = 9b+9 N=8c+7: N+1 = 8c+8 N=7d+6 and so on…

Q: In the expansion of (2x+1)n , the coefficient of the x2 term is 40n , where n is a positive integer.…

A: Given (2x+1)^n The solution is given below in the next step

Q: If x is an odd integer then 5x – 7 is an even integer.

A: 5 and x are odd integers. Multiplication of two odd integers is always odd. Hence, 5x is an odd…

Q: Find a formula for the number of digits of 2n.  

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Q: In a sports league of x teams in which each team plays every other team twice, the total number N of…

A: Since you have asked multiple questions in a single request, we would be answering only the first…

Q: Suppose, and no are the smallest of two positive integers so that 300x is a perfect square and 300y…

A: The given problem is to find the values of x and y and sum of value of x+y for which smallest…

Q: Show that n22-1 is division by 8, if n is an odd positive integer

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Q: A student has scores of 78, 77.25, and 81.75 on his first three tests. He needs an average of at…

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Q: Find a simple closed expression for computing E=1k²Hx and verify your answer for n = 4.

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

A: we have to Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the…

Q: Show that n2-1 is division by 8, if n is an odd positive integer

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Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: Find the sum 1 9k2 + 12k – 5 k=1 Express your answer as a rational expression in terms of n.

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Q: How many integer solutions are there to x1+ x2 + x3 = 0  with Xi >= -5 ?

A: We need to find the integer solutions to x1+x2+x3=0 such that each of the three numbers is greater…

Q: Find the smallest integer n for which the property below hold 3n > 2967 n =

A: Given query is to find the value of n.

Q: nth term of an AP  1+n+n2

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Q: Find an expression, in terms of n for the coefficient of x in the expansion (1+4x) + (1+4x)² +…

A: ANSWER :

Q: If in the expansion of (1 + n}+", the 5th term is 24 times the 3rd term, fnd the value of n.

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Q: Find an integer that leaves a remainder of 2 whan divided by either 3 or 5, but that is divisible by…

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Q: Find the term of (a – 2b)3 that has b in it. |

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Q: Find all sollutions in integers of 34x+8y=6

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Q: Without expanding completely, find the indicated term(s) in the expansion of the expression.

A: The Binomial Theorem is, (a+b)n=∑k=0nnkan-kbk Where, nk=n!k!(n-k)! The given question is to find the…

Q: Find the indicated term of each 49. sixth term of (4h - j)8

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Q: perform the indicated operations involving rational numbers. Express final answers in reduced form.…

A: Given: 47÷16-21

Q: John found the sum of the first 2021 positive even integers and Bob found the sum of the first 2022…

A: In this question, tne concept of sum of first positive even integers is applied.

Q: Determine how many terms of (-1)*+12k k! must be summed to be sure that the remainder term k=1 is…

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Q: 3. Determine the nth-term formula for the number of square tiles in the nth figure. а. a2 a4 as b.…

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Q: How many positive integer solutions are there to x1 + x2 + x3 =15 ?

A: From the given information we want to find How many positive integer solutions are there to x1 + x2…

Q: Find all positive integers n for which n1º +1 is divisible by 10.

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Q: Express the following using factorials: 10 x 9 x 8 x 7

A: Meaning of factorial of a natural number : Suppose n is a natural number then meaning of n! is…

Q: In a round-robin chess tournament, each player is paired with every other player once. The formula 2…

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Q: Find integers p and q so that 17p + 45q = 1. Show your working steps.

A: Here we have to find the integers p and q so that  17p+45q=1

Q: How many zeros are there at the end of 1001?

A: This question is related to Number Theory

Q: Show that n22-1 is division by 8, if n is an odd positive integer

A: If n is a odd positive integer n can be written as   n = 2k+1 by substituting in n2-1 n2-1 =…

Q: Without expanding completely, find the indicated term(s) in the expansion of the expression.

A: The (r+1)th term of the expression (a+b)n is given by Crnan-rbr. Here C represents the combination.…

Q: ss the following as a factorial: 6 x 5 x 4 x 3x 2 x

A: The factorial function tells us to multiply all whole numbers down to 1 from the selected number.

Q: Suppose n is odd. Find the cube of n and divide it by 4. What possible remainders could occur? Check…

A: Solution

Q: Find and simplify: D¿[In(3!nx²-x)]

A: We have to calculate the value of Dx[3ln(x^2-x)]

Q: Evaluate the first 5 terms (starting with n = 0) of an = n! %3D Simplify so that ! does not appear…

A: Given: an=xnn! To find: First five terms starting with n = 0

Q: If 6th term of a G.P is 9375 and its 4th term is 375.Find its 9th term

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.