Question

The sequence starts with an index of 1 and is defined so that is the largest integer such that .

a) Determine the recursive formula for the sequence.

b) Find the explicit formula for the sequence.

c) Identify the limit of the sequence.

d) Calculate the sum of the first n terms of the sequence.

Answer 1

a) The recursive formula for the sequence is an = max(n, an-1 + n). b) The explicit formula for the sequence is an = n(n+1)/2. c) The limit of the sequence is infinity. d) The sum of the first n terms of the sequence is Sn = (n/2)(n+1).

Step-by-step explanation:

a) To find the recursive formula for the sequence, we can consider the pattern. The sequence starts with an index of 1, and each term is the largest integer such that the sum of all the terms up to that index is less than or equal to the index itself. So, the recursive formula for the sequence can be written as:

an = max(n, an-1 + n)

b) To find the explicit formula for the sequence, we can solve the recursive formula to express an in terms of n:

an = n + (n-1) + (n-2) + ... + 1

an = n(n+1)/2

So, the explicit formula for the sequence is an = n(n+1)/2.

c) The limit of the sequence can be found by taking the limit as n approaches infinity:

limn→∞ an = limn→∞ n(n+1)/2 = ∞

So, the limit of the sequence is infinity.

d) The sum of the first n terms of the sequence can be found using the explicit formula:

Sn = (n/2)(n+1)

Answer 2

The recursive formula for the sequence is a(1) = 1 and a(n) = a(n-1) + 2n+1. The explicit formula for the sequence is a(n) = n^2. The sequence is unbounded and does not have a limit. The sum of the first n terms of the sequence is given by the formula S(n) = n(n+1)(2n+1)/6.

Step-by-step explanation:

a) Recursive formula for the sequence:

The recursive formula is given by:

a(1) = 1

a(n) = a(n-1) + 2n+1

b) Explicit formula for the sequence:

The explicit formula is given by:

a(n) = n^2

c) Limit of the sequence:

The sequence is unbounded, which means it does not have a limit.

d) Sum of the first n terms of the sequence:

The sum of the first n terms is given by the formula:

S(n) = n(n+1)(2n+1)/6