Question

Find the next five terms, then write a conjecture 1.) 13,18,23,28
2.) 512,256,128,64

Answer 1

1) 13,18, 23, 28, 33, 38, 43, 48, 53

2)512, 216, 128, 64, 32, 16, 8, 4, 2

Explanation:

We have to find the succeeding five terms of the given series

1) The given series is 13,18, 23, 28,...

If we see the given series carefully we see that it is an arithmetic progression.

An arithmetic progression is of the form a, a+r, a+2r, a+3r, ...

In the given series a = 13 and r = 18 - 13 = 5

Thus, the succeeding term of the series will be given by adding 5 to the preceding term.

The next five terms are:

28 + 5 =33

33 + 5 = 38

38 + 5 = 43

43 + 5 = 48

48 + 5 = 53

2) The given series is 512, 216, 128, 64,...

If we see the given series carefully we see that it is an geometric progression.

An geometric progression is of the form
`a, ar, ar^2, ar^3, ar^4, ...`

In the given series a = 512 and r =
`(216)/(512) = (1)/(2)`

Thus, the succeeding term of the series will be given by multiplying
`(1)/(2)` to the preceding term.

The next five terms are given by:

`64*(1)/(2) = 32`

`32*(1)/(2) = 16`

`16*(1)/(2) = 8`

`8*(1)/(2) = 4`

`4*(1)/(2) = 2`

Answer 2

1) Next five terms are 33, 38, 43, 48, 53

Notice that the difference between consecutive terms is 5.

Conjecture: {13 + 5n}, n = 0, 1, 2, ...

2) Next five terms are 32, 16, 8, 4, 2

Notice that each term is half of its preceding term.

Conjecture: {
`2^(9-n)`}, n = 0, 1, 2, ...