Question

What is the rule or pattern for each sequence, and what are the next three terms fo the sequences? 1. 6, 7, 9, 12,... 2. 28, 24, 20, 16,...

what could be the formula for the 2 question?
Ex) `a_n=3+4(n-1)`

I know the formula that adds the same number constantly but I don't know the rest.

Answer 1

Please see below.

Explanation:

.

`color(red)(1))`

`6,7,9,12,...`

`a_0=6+0=6`

`a_1=7=a_0+1=6+1`

`a_2=9=a_1+2=6+1+2`

`a_3=12=a_2+3=6+1+2+3`

This pattern shows that each term is equal to the first term plus the sum of a group of integers whose number is equal to the index of the term.

As can be seen, `a_2` is equal to `6` plus the sum of `2` integers starting with `1`. Similarly, `a_3` is equal to `6` plus the sum of `3` integers starting with `1`, and so on.

Therefore:

`a_n=6+(1+2+3+4+.....+n)`

But we know that the formula for the sum of integers is:

`(n(n+1))/2`

As such, the formula for this sequence would be:

`a_n=6+(n(n+1))/2`

`color(red)(2))`

`28, 24, 20. 16,....`

`a_0=28`

`a_1=24=a_0-4`

`a_2=20=a_1-4=a_0-4-4`

`a_3=16=a_2-4=a_0-4-4-4`

This shows the pattern that each term is equal to `a_0` minus a number that is equal to the index of the term multiplied by `4`.

`a_0=a_0-(0)(4)=28-0=28`

`a_1=a_0-(1)(4)=28-4=24`

`a_2=a_0-(2)(4)=28-8=20`

`a_3=a_0-(3)(4)=28-12=16`

Therefore, the formula for the sequence is:

`a_n=28-4n`