1.7 Show that 1² +2²+ +n² -1 n(n +1) (2n + 1).

Q: (5) 7(3k – 1)² +26(3k – 1)–8

A: On simplifying the above expression,we get

Q: Show that P(n, 1) + P(n, 2) = (22) – 2 (2), for integers n ≥ 2.

A:

Q: (2n)! – (2n + 1)! (2n)!

A: It is required to find out , 2n! - 2n+1!2n!  For this we use factorial properties.

Q: Differentiate f (x) = 71n(x²-6x+3) A. (2x – 6) 7ln(x²-6x+3) f' (x) = x2 – 6x + 3 - В. In(7) (2x – 6)…

A: Given query is to find the derivative of the expression.

Q: How many strings of 1our decimal digits have all four digits that are different? How many strings of…

A:

Q: { :Find the first 3 terms for 2n-1 (2n-1)! (−1)n+¹

A: given sequence  -1n+1x2n-12n-1!

Q: Let a, = 1 and a₁ + a1 = n² + 2n, n ≥2. Evaluate S

A:

Q: 10. Use the binomial theorem to evaluate (2r³ - y) 5.

A:

Q: 21 divides 4(n+1) + 5(2n-1) whenever n is a positive integer.

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

Q: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

A:

Q: Which simplified choice will result in 200m20n15

A: Consider the following expression…

Q: Which of the following is equivalent to 3x²(-2x)e-x² + e-x²16x O -16xe-x²(x² - 1) O 16xe-x²(x² + 1)…

A: Given query is to simplify the expression.

Q: "Co 34 "C, "C, "C; (-1)" "C, ,ne Nis equal to 10 14 (4n + 2) 2" n! 2"-n! () 3.5-7...(2n - 1)(2n +1)…

A: We have use binomial theorem  1-xn=nC0 - nC1 x + nC2 x2  + .........+(-1)n   nCn

Q: 3) (n² + 10n+18) ÷ (n+ 5)

A: To divide the following expression.n2+10n+18÷n+5

Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: 3. Determine the 4th term of (3x+ 2y3 6 using The Binomial Theorem.

A: We will find the 4th term

Q: 20. (a) Fill in the blank: 1+3+5+..+ (2n – 1) = k=1 (b) Use part (a) to prove that the sum of the…

A: This is an arithmetic series with common difference (3-1)=2 And this has n terms in it So we use the…

Q: Indicate the least integer n such that (3x5 + x²)4 = 0(x¹). E

A:

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: use gauss's trick to determine R(n)=2+3+4+...+(n-1)+n+(n+1)

A: It is known that sum of first n numbers is n(n+1)2  that is  1+2+3+----+n=n(n+1)2 Gauss trick method…

Q: 2021 2. Expand (1 + v3i)“ recall the binomial theorem.

A:

Q: ndicate the least integer n for which (3x5 + x2)4 is O(xn).

A: Solved using the binomial theorem

Q: (b) Show that 1 11 2 k(k + 2)(k + 4) 96 k=1

A: Given series is ∑k=1∞1kk+2k+4

Q: Verify that for all n ≥ 1, the sum of the squares of the first 2n positive integers equals…

A:

Q: Part 1. Obtain a solution to the following differential equation valid near the origin: 1.] For the…

A: Part 1. Given equation is y''+3xy'+7y=0   4. To Find: The recurrence relation for n≥2   5. To Find:…

Q: a) 1 (x-1)(x+2)(x+4)

A:

Q: (Vn)(n2 +n+ 17 is prime)

A: Since, you have posted multiple questions , I am doing the very question (ques 7)  according to our…

Q: 2. Find (w + 1)(w-w+1).

A:

Q: (b) Prove that 32n²+1, for n ≥ 1.

A: The given inequality is 3n≥2n2+1, for n≥1. We have to prove this inequality.