Quadratic Sequence and Geometric Sequence

In a geometric sequence, divide each term by the term before it to find the common ratio $r$.

$10÷5=2$, $20÷10=2$ and $40÷20=2$.

The ratio is the same every time, so the common ratio is 2.

Find the common ratio by dividing a term by the one before it: $32÷64=0.5$.

Check it is the same all the way along: $16÷32=0.5$ and $8÷16=0.5$, so the ratio really is constant at $0.5$.

The ratio is less than 1, so each term is half the one before and the terms get smaller.

Multiply the last term by the ratio: $8\times 0.5=4$.

So the next term is 4.

A quadratic sequence has a constant second difference, so start with the first differences.

$4-3=1$, $7-4=3$, $12-7=5$, $19-12=7$, giving the row $1,3,5,7$.

Now difference that row: $3-1=2$, $5-3=2$, $7-5=2$.

So the second difference is constant at 2.

Second difference means the difference between the first differences. It is not one of the first differences itself, and it is not the gap between terms two places apart.

First find the first differences: $8-5=3$, $14-8=6$, $23-14=9$, $35-23=12$.

These are not constant, so difference them again: $6-3=3$, $9-6=3$, $12-9=3$. The second difference is 3.

Because the gaps grow by 3 each time, the next first difference is $12+3=15$.

Add that to the last term: $35+15=50$.

So the next term is 50. Note that the second difference is added to the gap, not to the term itself.

The nth term of a geometric sequence is $a\times r^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

Here $a=3$, $r=2$ and $n=6$, so the 6th term is $3\times 2^{6-1}=3\times 2^5$.

$2^5=2\times 2\times 2\times 2\times 2=32$, so $3\times 32=96$.

So the 6th term is 96.

You can check by listing the terms: 3, 6, 12, 24, 48, 96. Getting from the 1st term to the 6th takes five multiplications, which is why the exponent is $n-1$ and not $n$.

Find the first differences: $7-4=3$, $14-7=7$, $25-14=11$, $40-25=15$.

Difference that row again: $7-3=4$, $11-7=4$, $15-11=4$, so the second difference is constant at 4.

The gap grows by 4 every time. The next gap is $15+4=19$, so the 6th term is $40+19=59$.

The gap after that is $19+4=23$, so the 7th term is $59+23=82$.

So the 7th term is 82. The gap must be increased again for the second step - it does not stay at 19.