Sequence Problems

Arithmetic: Sequence Problems

Read the following SAT test question and then select the answer.

Always read the question carefully and identify the bottom line.  Then assess your options and use the most efficient method to attack the problem.  When you have an answer, loop back to make sure that you solved for the bottom line.

8, a, 14, b, 20, …
The first term of the sequence above is 8. Which of the following could be the formula for finding the nth term of this sequence for any positive integer n?

Bottom Line: You want a formula to describe this number sequence.

Assess your Options:  You could try to write a formula, but you will have a hard time doing that because you do not know the second and fourth terms in your pattern.  You also do not need to find numbers for the variables a and b in order to solve this problem.  Instead, use the answer choices to help you find an answer.

Attack the Problem: The first thing to do is realize that n is not a variable that you have to find algebraically; the nth term just describes the number of that term in the sequence, like the first, second, third, fourth, or fifth.  Therefore:
8,   a,  14,  b,   20, …
1,   2,   3,   4,     5

That means that when you plug in 1 to the formula, you should always get 8, when you plug in 3, you should always get 14, and when you plug in 5, you should always get 20.

Take a look at your answer choices:

(A) 2n + 6
(B) 3n + 5
(C) 5n + 3
(D) 6n + 2
(E) 6n + 5

You could start by plugging in 1 and finding out which of these equals 8, eliminate any that do not, and then try plugging in 3 and then 5 (this method is used on collegeboard.org).  However, just by looking at the numbers (a lot of 2s and 6s and a lot of 3s and 5s) you should be able to tell that a lot of these will equal 8.  To save time, start by plugging in the biggest term you know, the fifth, and see which answer choices equal 20.

(A) 2(5) + 6 = 16
(B) 3(5) + 5 = 20
(C) 5(5) + 3 = 28
(D) 6(5) + 2 = 32
(E) 6(5) + 5 = 35

Note: if you use logic, you do not even have to work out (C), (D), and (E) because the product of the first two numbers is larger than 20 before you even add to them.

Only one answer choice results in the correct 5th term of 20.  You don’t need to check any other numbers!

Loop back:  You found the only formula that will work for every number in the sequence, so select that answer.

(A) 2n + 6
(B) 3n + 5
(C) 5n + 3
(D) 6n + 2
(E) 6n + 5

The correct answer is (B).

On sat.collegeboard.org, 43% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Sequences

Link of the Day

Taking the SAT (or any test for that matter) can be very stressful. It can be difficult to relax and focus under pressure. In fact, often once you start panicking things only get worse. There are lots of different ways to manage test day stress. It's important to make sure that you get a good night sleep, eat a good breakfast (though you should avoid a really heavy breakfast), and arrive early at the testing site. You can find some great suggestions on how to minimize your test day stress here

6/11 Sequences

Always follow the Knowsys Method for math problems. It will save you time not only on the SAT, but also on math tests and quizzes in your school classes. Read the question carefully and identify the bottom line. Assess your options. Ask what you could do and then what you should do, and solve the problem quickly and efficiently once you have decided on a strategy. Finally, loop back to double check that you answered the question correctly.

8, a, 14, b, 20, ...

The first term of the sequence above is 8. Which of the following could be the formula for finding the nth term of this sequence for any positive integer n?

This problem really demonstrates how important it is to follow the Knowsys method (and to not skip steps). After reading the problem carefully and identifying the bottom line (you are looking for the formula for the nth term of this sequence), you may think that this requires the use of some complicated formula. If you are panicking right now because you can't remember what a geometric sequence is, or how to solve one, take a deep breath and relax. This problem is actually much simpler than that. Remember that step three of the Knowsys method is to assess your options. Ask yourself what could you do, and what should you do. You could try to solve this problem by writing a formula for the sequence, or you could simply work backwards and check the answers. You should solve this problem by working backwards because it is much easier and much faster.

(A) 2n + 6
(B) 3n + 5
(C) 5n + 3
(D) 6n + 2
(E) 6n + 5

We know that the first term in the sequence (when n = 1) must be 8, the third term in the sequence (when n = 3) must be 14, and the fifth term in the sequence (when n = 5) must be 20. Now all you have to do is try each formal and find the correct answer.

(A) works for the first term (since 2 * 1 + 6 = 8) but it fails for the third term (2 * 3 + 6 = 12, not 14).
(B) works for the first term (3 * 1 + 5 = 8), the third term (3 * 3 + 5 = 14), and the fifth term (3 * 5 + 5 = 20). Don't forget to loop back and verify that your answer matches the bottom line (it does). You now know this is the correct answer so you should simply bubble in (B) and move on from here.

For the sake of being thorough, here is an explanation of why the other answers do not work.

(C) works for the first term (5 * 1 + 3 = 8) but fails for the third term (5 * 3 + 3 = 18, not 14)
(D) works for the first term (6 * 1 + 2 = 8) but fails for the third term (6 * 3 + 2 = 20)
(E) fails for the first term (6 * 1 + 5 = 11)

The correct answer choice is (B).

On sat.collegeboard.org 46% of the responses were correct.

For more help with math, visit www.myknowsys.com