SAT Sequences and Series

Correct Answer: B

Solution 1:

$8$ is obtained by multiplying the preceding term, $6$ , by $x$ and then subtracting $y$ . Translating the words into math, $8=6x-y$ . Similarly, $12$ is obtained by multiplying the preceding term, $8$ , by $x$ and then subtracting $y$ . This means that $12=8x-y$ . We use the first equation to solve for $y$ in terms of $x$ . Then we substitute for $y$ in the second equation.

$\begin{array}{l c l l l} 8&=&6x-y &\quad \text{expression for second term in sequence} &(1)\\ 8-6x&=&-y &\quad \text{subtract}\ 6x\ \text{from both sides} &(2)\\ 6x-8&=&y &\quad \text{divide both sides by}\ -1 &(3)\\ \end{array}$

Substituting for $y$ in the second equation, we get:

$\begin{array}{l c l l l} 12&=&8x-y &\quad \text{expression for third term in sequence} &(4)\\ 12&=&8x-(6x-8) &\quad \text{substitute}\ 6x-8\ \text{for}\ y &(5)\\ 12&=&8x-6x+8 &\quad \text{use distributive property} &(6)\\ 12&=&2x+8 &\quad \text{combine like terms} &(7)\\ 4&=&2x &\quad \text{subtract}\ 8\ \text{from both sides} &(8)\\ 2&=&x &\quad \text{divide both sides by}\ 2 &(9)\\ \end{array}$

Now that we know that $x=2$ , we can solve for $y$ :

$y=6x-8=6\cdot2-8=12-8=4$ .

Solution 2:

We obtain expressions for the second and third term of the sequence. $8$ is obtained by multiplying the preceding term, $6$ , by $x$ and then subtracting $y$ . So, $8=6x-y$ . Similarly, $12$ is obtained by multiplying the preceding term, $8$ by $x$ and then subtracting $y$ . This means that $12=8x-y$ . We have the two equations:

$\begin{array}{l c l l l} 8&=&6x-y &(1)\\ 12&=&8x-y &(2)\\ \end{array}$

We subtract $(1)$ from $(2)$ , and we get:

$\begin{array}{l c l l l} 12-8&=&(8x-y)-(6x-y) &\quad \text{subtract}\ (1)\ \text{from}\ (2)\ &(3)\\ 4&=&8x-y-6x+y &\quad \text{use distributive property} &(4)\\ 4&=&2x &\quad \text{combine like terms} &(5)\\ 2&=&x &\quad \text{divide both sides by}\ 2 &(6)\\ \end{array}$

Now that we know that $x=2$ , we can solve for $y$ :

$y=6x-8=6\cdot2-8=12-8=4$ .

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Incorrect Choices:

(A)
Tip: Read the entire question carefully.
If you solve for $x$ , instead of $y$ , you will get this wrong answer.

(C)
Tip: Read the entire question carefully.
You may notice that we can obtain each successive term of the sequence by adding to the preceding term a power of two:

$\begin{array}{lll} 6 &=&6+2^{0}&\text{first term}\\ 8&=& 6+2^{1} &\text{second term}\\ 12&=&8+2^{2} &\text{third term}\\ 20&=&12+2^{3} &\text{fourth term}\\ \vdots &&& \vdots\\ \end{array}$

If you answer the question, "What must be added to the fourth term of the sequence to obtain the fifth term," you will get this wrong answer.

(D)
If you solve for the next term in the sequence, instead of solving for $y$ , you will get this wrong answer.

(E)
You may get this wrong answer if in step $(6)$ of Solution 1 you forget to distribute the negative sign, like this:

$\begin{array}{l c l l l} 12&=&8x-(6x-8) &\quad &(5)\\ 12&=&8x-6x\fbox{-}8 &\quad \text{mistake: didn't distribute negative sign} &(6)\\ \end{array}$