What is the coefficient of x7 in (3-x)10 O a. C(10,7)* -37 O b. C(10,3)* 3³ O c. P(10,7)* -33 O d. -C(10,3)* 33

Q: 15) n(n*+5) is divisible by for each integer

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Q: Can you use mathematical induction to show that 4 divides 32x + 7 First define the predicate ?(x)

A: given , expression :- 32x+7 we have to prove that given expression is divisible by 4 .

Q: Let 4 a) Given that In (y) can be written in the form In (v) = In (A) =³ + In (B) =² + In (C) z+ In…

A: The given exponetial function is: y=4x2-4x ·5x3-5x2·72x+3 Now, taking the log on both sides.…

Q: 12 Find the term of (a – 3b)" that has b in it.

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Q: Can you use mathematical induction to show that 4 divides 32x + 7 First define the predicate P(x)

A: Yes we can use mathematical induction for Natural number x.

Q: What is the coefficient of x' in (3-x)10

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Q: What are the values for x1, x2 and x3? O x1-500 ; x2-57.78, x3-175.56 O x1-500; x2=-175.56, x3-57.78…

A: We have given optimization table and we have to find the values of variables that maximize the…

Q: (a) In (x - 3) = 7 (b) In (x + 4) + In (x- 4) = 1

A: Given(a) In(x-3) = 7(b) In(x+4) +In(x-4) =1solve for x(a) In(x-3) = 7In(x-3) =7take Ine  both…

Q: show that n(n^2-1)(n+2) is divisible by 4 for all integers n

A: To show that n(n^2-1)(n+2) is divisible by 4 for all integers n

Q: The coefficient of z in the expansic

A: Given expression is  1-coszz. We have to find the coefficient of z5 in 1-coszz about z=0.

Q: The factorial of n is the product of all positive integers less than or equal to n. It is…

A: Note:Hey there! Thank you for the question! We have solved this problem using trial-and-error…

Q: (b) Suppose a, b, and c are integers and suppose a | b and a{c, where a{c denotes that c is not…

A: Given, (i) a divides b  (ii) a doesn't divide c  To show : a doesn't divide (b+c) On the contrary,…

Q: Find all integers x + 3 such that x – 3 is divisible by x – 3. |

A: To find all integers such that x^3-3 is divisible by x-3

Q: 2/ Let the Polf fext = then E(X) is 2. 225 (30-x) 225 ③2x815 15 ≤x≤ 30

A: It is given that  The pdf of X is, f(x) = {  x/225,   0≤x≤15              (30-x)/225,   15≤x≤30 }.

Q: Find all four-digit numbers abba with the property aa . 10b= abba if a is not equal to b.

A: We know if abcd is a four - digit number then, abcd = 1000a+100b+10c+d

Q: Use the Principie to prove the Ving 3n+1 – 1 is divisible by 2 for all n > 0

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Q: Vn E N, nº – n is divisible by 6.

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Q: True or False: There are integers x, y such that 15 = 215x + 497y. O True O False

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Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: 10. If a, b and p are positive integers, what is the LCM of ap, bp^3 and a^2p^2? 1 a^2bp^3 abp p^2

A: Since the student has mentioned to solve multiple questions (no. 10-12), but we can solve one…

Q: what is the coefficient of X^9 in (x+3)^13

A: An algebric expression of two number connected by operations "+" or "-" is called binomial. The…

Q: Which of the following is a coefficient of x' in (2 - x)11, O -5280 O 5280 О -330 О 330

A: In the expansion of (a+b)n , the (r+1)th term is given by tr+1 = ncr.an-r.br

Q: 11. What do you obtain when you raise 27 to the 8/3 power ? * 1/3 of (27 to the 8th power)

A: Any number a can be written as , where b is any natural number. Also, 27 is a cube of 3, so 27 can…

Q: What is the coefficient of z12 in (6-12z)12 O a. P(12,6)* -612 O b. C(12,0)* -(1212) O . 1212 O d.…

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Q: - Show that an = is monotone. n+2n+7 "}= •

A:

Q: 3. Show that 13" – 7ª is divisible by 6 for n > 1.

A: As per Bartleby guidelines we can solve any one question at a time. Hope you understand.

Q: 4. Let, a = 3, a2 = 4 and for n 2 3, a, = 2an-1+ an-2 +n, express a, in terms of n.

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Q: Suppose m = apɑp-1*** a2ɑ1ao and n = bpbp-1*…* b½b¡bo are two numbers with the same digits so that n…

A: 7) a. Given that ai=bj. Without loss of generality assume that i>j. Now the value of the digit ai…

Q: Find a formula for the general term a, {5, 8, 11, 14, 17, .} 5n + 3 On+3 O 3n + 2 O 2n + 3

A: We can answer the question as below

Q: 2. Show that if a]b then a'|b² for all integers a and b (Symbols V, 3, A, V, V,A,→,-)

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Q: What is the row of Pascal's triangle containing the binomial coefficients (), 0 < k < 9?

A: The first row of pascal triangle contains only one element i.e. 1. In second row, there are two…

Q: What is the coefficient of p5q5 in the expansion of (p+q)10?

A:

Q: Let a be an integer such that the remainder of dividing a by 17 is 15. What is the remainder of…

A: We will find out the required value.

Q: Find the coefficient of x9 in (2-x)19.

A:

Q: How many integer solutions to x1+ x2+ x3+ x4 = 36 are there for which æ1 > 5, x2 > 3, x3 > 5 and æ4…

A: We have to find the number of solutions to x1+x2+x3+x4=36 for which x1≥5, x2≥3, x3≥5 &amp; x4≥6.

Q: What are the 5a + 20a - 60 factors of O 5 (x-6)(x+2) • 5 (x+6)(x-2) O 5 (x-6)(x-2)

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Q: (d))For all integers a, b, and c, if a | b or a c, then a | bc (e) For all integers a, b, and c, if…

A: (d) let for all integers a,b,c if a|b or a|c so a|b or a|c ⇔b = ak1 or  c = ak2  for some integers…

Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

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Q: The n in X" indicating the number of factors of X.

A: solution:

Q: 5. Find the value of AB. B AL 5 in 5 in X16 in

A: We have to find AB

Q: 2. 6n – 1 is divisible by 5 for n any positive integer

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Q: Obtain the expansion of : 2 up to the term in x8 1-3x2

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Q: What is the coefficient of æ10 in (æ + 3)13?

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Q: Find the coefficient specified in each item. a) a4 in (6−1⁄2a)6

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Q: Is it true that for any odd n the number 2" +1 is divisible by 3? Prove your answer.

A: We prove this by using PMI(Principle of Mathematical induction)

Q: The greatest common divisor of 9n + 10 and 10n +9 is Oa 10 or 19 оь. 19 O. None d. 1 or 19 Oe. 1

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Q: How do you show that x3 is O(x4) but x4 is not O(x3)?

A: Recall the definition of Big-O notation which states that fx is Ogx if there exist constants C and k…

Q: Prove that n(n + 1)(n + 2) is always divisible by 6

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Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: