This probably easy asf for y’all but I’m doing GCSEs and this question is baffling me

[steady needle]

Im looking at the answer and working and have no clue

how they are getting the numbers

[unborn marlin]

@steady needle you have a series with 3 variables, namely, a, b, and n. n itself is the input variable, or in other words, the variable you would put the number into to get the nth term.

You are given that when n = 2 the term equals -2, and you are given when n = 4 the term equals 12. You can solve for a and b using a system of equations, and from there find the 6th term of the series.

[normal bone]

Given:4a+2b=-2--(1)
16a+4b=12--(2)
Wanted: 36a+6b=?
so equation one is
4a+2b=-2 divide both sides by 2
2a+b=-1
equation two is
16a+4b=12 divide both sides by 4
4a+b=3
subtracting them gives
2a=4=>a=2
put a=2 in eqn 1
2(2)+b=-1=>b=-5
verify
4(2)-5=3=>3=3
so values for and b are correct
36(2)+6(-5)
72-30=42

hope it helps

[hazy smelt]

so the nth term of the sequence is $an^2 + bn$

[hoary nest]

iaminfinityiq

[hazy smelt]

so the formula for the 2nd term of the sequence should be $a \cdot 2^2 + b \cdot 2$

[hoary nest]

iaminfinityiq

[hazy smelt]

bc we substitute 2 in

right?

so that's $4a + 2b = -2$ since the second term is -2

[hoary nest]

iaminfinityiq

[hazy smelt]

do the same thing for the 4th term, we should get $16a + 4b = 12$

[hoary nest]

iaminfinityiq

[hazy smelt]

in this situation

we can eliminate the $a$ term or eliminate the $b$ term

[hoary nest]

iaminfinityiq

[hazy smelt]

by doing some stuffs with the 2 equations by multiplying one or 2 equations by different numbers

in this case, we can eliminate the $b$ term

[hoary nest]

iaminfinityiq

[hazy smelt]

by doubling the first equation

we should get $8a + 4b = -4$

[hoary nest]

iaminfinityiq

[hazy smelt]

and since $16a + 4b = 12$

[hoary nest]

iaminfinityiq

[hazy smelt]

we can take the second equation, subtracting the first equation

or $(16a + 4b) - (8a + 4b) = 8a = 12 - (-4) = 16$

[hoary nest]

iaminfinityiq

[hazy smelt]

from that, we should get $a = 2$

[hoary nest]

iaminfinityiq

[hazy smelt]

sub that in the first equation, we get $4 \cdot 2 + 2b = -2$

[hoary nest]

iaminfinityiq

[hazy smelt]

or $8 + 2b = -2$

[hoary nest]

iaminfinityiq

[hazy smelt]

$2b = -10$ or $b = -5$

[hoary nest]

iaminfinityiq

[hazy smelt]

so now

we got the formula for the nth term is $2n^2 - 5n$

[hoary nest]

iaminfinityiq

[hazy smelt]

now the next questions should be easy now...