Consider a sequence given by the formula an = an−1 − 5, where a1 = 12 and n ≥ 2.
c. find a6 and a100 of the sequence

a6 = -13

a100 = -507

Step-by-step explanation:

To find the value for a6 and a100 we first need an explicit formula for the sequence

To find the explicit formula we need to find the first few terms of the sequence (in this case we take the first 5)

To find the first 5 terms we have the initial value and using the given formula we find the remaining 4 values

the 5 values are as follow

a1 = 12

a2 = 7

a3 = 2

a4 = -3

a5 = -8

From this we know the initial value which is 12 and the constant difference between each value which is -5

Using the general formula for arithmetic sequences i.e.

where

is the nth term of the sequence

is the initial value which is 12

is the nth count

is the constant difference between each term which is -5

Thus we have the following formula

Finding the a6 or 6th value

Finding the a100 or 100th value

51 and 42

Step-by-step explanation:

Use

x + y = 93

x - y = 9

Add those two equations together and you get 2x = 102

*y is not in the equation anymore because y + -y = 0*

Divide both sides by 2

x = 51

plug x into one of the equations; let's use x + y = 93

Then you get, 51 + y = 93

Subtract both sides by 51 and you get y = 42

Plug both in the other equation to make sure the numbers work.

51 + 42 = 93; true

51 - 42 = 9; true