Question

Find the nth term of a sequence whose first several terms are given.
5,
11
4
,
17
9
,
23
16
,
29
25
, . . .
an =

Answer 1

The sequence appears to be following a pattern where the even-numbered terms increase by 6 each time, starting from 4, and the odd-numbered terms increase by 8 each time, starting from 5.

If we break down the pattern:

- The even-numbered terms increase by 6: 4 + 6 = 10, 10 + 6 = 16, 16 + 6 = 22, ...
- The odd-numbered terms increase by 8: 5 + 8 = 13, 13 + 8 = 21, 21 + 8 = 29, ...

From this pattern, we can see that the even terms are following the arithmetic sequence: 4, 10, 16, 22, ... with a common difference of 6, and the odd terms are following the arithmetic sequence: 5, 13, 21, 29, ... with a common difference of 8.

Now, we need to identify the patterns for the position of the term in the sequence. We can notice that the position of the term corresponds to the term number itself:

- 1st term: 1
- 2nd term: 2
- 3rd term: 3
- 4th term: 4

From this, we can see that the position of the term is equal to the term number itself.

To find the nth term, we first need to determine whether the term is even or odd. We can do this by finding the remainder when dividing the term number (n) by 2. If the remainder is 0, the term is even. If the remainder is 1, the term is odd.

If the term is even, we use the formula for the nth term of an arithmetic sequence to find the value:

Even terms: an = a1 + (n/2 - 1)d

- a1 = 4 (the first term of the even sequence)
- d = 6 (the common difference for the even sequence)

If the term is odd, we use the same formula but with the odd sequence:

Odd terms: an = a1 + (n/2 - 1)d

- a1 = 5 (the first term of the odd sequence)
- d = 8 (the common difference for the odd sequence)

Let's substitute these values into the formula and calculate the nth terms.

For even terms (when n is even):

an = 4 + (n/2 - 1) * 6

For odd terms (when n is odd):

an = 5 + (n/2 - 1) * 8

For example:
- To find the 6th term of the sequence, we plug in n=6 into the formula for even terms:
a6 = 4 + (6/2 - 1) * 6
a6 = 4 + (3 - 1) * 6
a6 = 4 + 2 * 6
a6 = 4 + 12
a6 = 16

- To find the 7th term of the sequence, we plug in n=7 into the formula for odd terms:
a7 = 5 + (7/2 - 1) * 8
a7 = 5 + (3.5 - 1) * 8
a7 = 5 + 2.5 * 8
a7 = 5 + 20
a7 = 25

Therefore, an = 16 for even terms and an = 25 for odd terms.