Question

Suppose for each positive integer n, an is an integer such that a1 = 1 and ak = 2ak−1 + 1 for each integer k ≥ 2. Guess a simple expression involving n that evaluates an for each positive integer n. Prove that your guess works for each n ≥ 1.

Suppose for each positive integer n, an is an integer such that a1 = 7 and ak = 2ak−1 + 1 for each integer k ≥ 2. Guess a simple expression involving n that evaluates an for each positive integer n. Prove that your guess works for each n ≥ 1.

Answer #1

Here given a1=1 and ak=2a*k-1+1 ........eq(1) for
each integer* k≥2 .........eq(2)

n is positive integer.

Put k=2,3,4.... We get,

a2 = 2a2-1+1 = 2a1+1 = 2×1+1 = 2+1 = 3

(given a1=1)

Similarly,

a3=2a2+1=2×3+1=7

a4=2a3+1=2×7+1=15

So on..

Now we guess an expression involving n by the help of above expression which evaluate an for each positive integer i.e. n≥1.

Put k=n+1 in equation (2).

n+1≥2

=> n≥2-1

=> n≥1

See here if we put k= n+1 we get an expression that evaluates for each positive integer.

Put k=n+1 in equation (1), we get

an+1=2an+1-1+1=2an+1

an+1=2an+1 **is** **an**
**expression**.

Proof: put n=1,2,3,4....we get,

a1+1 = 2a1+1 = 2×1+1 = 3 (a1=1 given)

a2=3

Similarly,

a3=2a2+1=2×3+1=7

a4=2a3+1=2×7+1=15 hence proved.

2. Exercise 19 section 5.4. Suppose that a1, a2, a3, …. Is a
sequence defined as follows:
a1=1 ak=2a⌊k/2⌋ for every integer k>=2.
Prove that an <= n for each integer n >=1.
plzz send with all the step

let's fix a positive integer n. for a nonnegative integer k, let
ak be the number of ways to distribute k
indistinguishable balls into n distinguishable bins so that an even
number of balls are placed in each bin (allowing empty bins). The
generating function for sequence ak is given as 1/F(x).
Find F(x).

. Consider the sequence defined recursively as a0 = 5, a1 = 16
and ak = 7ak−1 − 10ak−2 for all integers k ≥ 2. Prove that an = 3 ·
2 n + 2 · 5 n for each integer n ≥ 0

Prove that for each positive integer n, (n+1)(n+2)...(2n) is
divisible by 2^n

1) Suppose a1, a2, a3, ... is a sequence of integers such that
a1 =1/16 and an = 4an−1. Guess a formula for an and prove that your
guess is correct.
2) Show that given 5 integer numbers, you can always find two of
the numbers whose difference will be a multiple of 4.
3) Four cats and five mice form a row. In how many ways can they
form the row if the mice are always together?
Please help...

Suppose p is a positive prime integer and k is an integer
satisfying 1 ≤ k ≤ p − 1. Prove that p divides p!/ (k! (p-k)!).

1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . . .
, ak) = 1, i.e., the largest
positive integer dividing all of a1, . . . , ak is 1. Prove that
the equation
a1u1 + a2u2 + · · · + akuk = 1
has a solution in integers u1, u2, . . . , uk. (Hint. Repeatedly
apply the extended Euclidean
algorithm, Theorem 1.11. You may find it easier to prove...

Discrete Math
6. Prove that for all positive integer n, there exists an even
positive integer k such that
n < k + 3 ≤ n + 2
. (You can use that facts without proof that even plus even is
even or/and even plus odd is odd.)

Find positive numbers n and a1
,a2,...,an such that
a1 + . . . an = 1000 and the product
a1 a2 . . . is as large as possible. Also
prove why?

Let λ be a positive irrational real number. If n is a positive
integer, choose by the Archimedean Property an integer k such that
kλ ≤ n < (k + 1)λ. Let φ(n) = n − kλ. Prove that the set of all
φ(n), n > 0, is dense in the interval [0, λ]. (Hint: Examine the
proof of the density of the rationals in the reals.)

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