A sequence a1, a2, a3,….. is defined by An+1 4аn — 3, п > 1 -- = k, where k is a positive integer. Given that E3_ a, = 66, find the value of k.

Q: Use Mathematical induction to show that H sub(2^n) is less than or equal to 1 + n.

A:

Q: A sequence {a„}1 satisfies a1 = 1, a2 = 1 and 6 an = max {an-1, an-2} – min {an-1, an-2} for n > 3.…

A:

Q: Show for all natural numbers n >= 3 that [(1 + (1/n))^n] < n.

A:

Q: 1. Write the first five terms of the sequence п а) а — 3" п b) а, = sin п c) a̟ = (- 1)'(÷) (–…

A:

Q: 5. A bit string is a sequence of numbers consisting of 0’s and 1’s. Answer the following questions…

A: (a) To fill each of the 5 places of a bit string of length 5, we have two choices (0/1).  So number…

Q: -) Determine the first six terms of the sequence defined by t₁ = -5 and t = 3t-1 + 8. 5, 23, -61,…

A:

Q: For a natural number n, we define n! = 1-2-3 (n − 1). n. - For example: 2! = 1-2, 3! = 1-2-3 = 6,…

A:

Q: Is there a way to do just using the starting value 2, instead of 2 and 5? In other words, can I get…

A: I am going to solve the problem by using some simple algebra to get the required result of the given…

Q: Find all integers n such that (n^2) - 1 is prime.

A: n² - 1 = (n-1)(n+1)  Now, if (n² - 1) needs to be a prime, then, either n-1 is equal to 1 or, n+1 is…

Q: 1 B. E 2n +1 2n +3 n=1

A: Solution of the above series is

Q: bers {1, 3, 6, …}

A:

Q: (-1)"n An n +1

A: We have to evaluate the limit of the sequence an=-1nnn+1.

Q: A sequence bo, b1, b2, ... is defined by letting bo = 5 and b = 4+bk-1 for each integer k> 1. Show…

A:

Q: Suppose An = 2n^2 + 6n + 6. Find the closed formula for the sequence of differences by computing An…

A: As per our question, we need to find the closed form of an-an-1×an-an-1 First find an-1 and then…

Q: (4) For all natural numbers n, 4^n - 1 is a multiple of 3

A:

Q: Show that n! > 3ª for n > 7.

A: The given inequality is, n!&gt;3n We need to prove that, the inequality is true for n≥7 Firstly,…

Q: A sequence can be generated by using a_(n)=a_((n-1))+4. If a_(1)=2, what is the value of a_(20).

A:

Q: Prove that 2^(27) - 1 is not a prime number.

A: A prime number is a natural number greater than 1 that is not a product of two smaller natural…

Q: How many ternary ( 0,1,2) sequences of length 10 are there without any pair of consecutive digits…

A: We see, we can start the desired ternary sequences with any of the 3 digits (0, 1 or 2).    Now, for…

Q: Find the 66th term of the triangular numbers {1, 3, 6, …}.

A: