### All PSAT Math Resources

## Example Questions

### Example Question #1 : Nth Term Of An Arithmetic Sequence

In an arithmetic sequence, each term is two greater than the one that precedes it. If the sum of the first five terms of the sequence is equal to the difference between the first and fifth terms, what is the tenth term of the sequence?

**Possible Answers:**

0.6

10

15.6

–0.6

2.4

**Correct answer:**

15.6

Let a_{1} represent the first term of the sequence and a_{n} represent the nth term.

We are told that each term is two greater than the term that precedes it. Thus, we can say that:

a_{2} = a_{1 }+ 2

a_{3} = a_{1} + 2 + 2 = a_{1} + 2(2)

a_{4} = a_{1} + 3(2)

a_{5} = a_{1} + 4(2)

a_{n} = a_{1} + (n-1)(2)

The problem tells us that the sum of the first five terms is equal to the difference between the fifth and first terms. Let's write an expression for the sum of the first five terms.

sum = a_{1} + (a_{1 }+ 2) + (a_{1} + 2(2)) + (a_{1} + 3(2)) + (a_{1} + 4(2))

= 5a_{1} + 2 + 4 + 6 + 8

= 5a_{1} + 20

Next, we want to write an expression for the difference between the fifth and first terms.

a_{5} - a_{1} = a_{1} + 4(2) – a_{1} = 8

Now, we set the two expressions equal and solve for a_{1}.

5a_{1} + 20 = 8

Subtract 20 from both sides.

5a_{1} = –12

a_{1} = –2.4.

The question ultimately asks us for the tenth term of the sequence. Now, that we have the first term, we can find the tenth term.

a_{10 }= a_{1} + (10 – 1)(2)

a_{10} = –2.4 + 9(2)

= 15.6

The answer is 15.6 .

### Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

In a certain sequence, *a*_{n+1} = (*a _{n}*)

^{2}– 1, where

*a*represents the

_{n}*n*th term in the sequence. If the third term is equal to the square of the first term, and all of the terms are positive, then what is the value of (

*a*

_{2})(

*a*

_{3})(

*a*

_{4})?

**Possible Answers:**

6

63

24

48

72

**Correct answer:**

48

Let *a*_{1} be the first term in the sequence. We can use the fact that *a*_{n+1} = (*a _{n}*)

^{2}– 1 in order to find expressions for the second and third terms of the sequence in terms of

*a*

_{1}.

*a*_{2} = (*a*_{1})^{2} – 1

*a*_{3} = (*a*_{2})^{2} – 1 = ((*a*_{1})^{2} – 1)^{2} – 1

We can use the fact that, in general, (*a –* *b*)^{2} = *a*^{2} – 2*ab* + *b*^{2} in order to simplify the expression for *a*_{3}.

*a*_{3 }= ((*a*_{1})^{2} – 1)^{2} – 1

= (*a*_{1})^{4} – 2(*a*_{1})^{2} + 1 – 1 = (*a*_{1})^{4} – 2(*a*_{1})^{2}

We are told that the third term is equal to the square of the first term.

*a*_{3} = (*a*_{1})^{2}

We can substitute (*a*_{1})^{4} – 2(*a*_{1})^{2 }for *a*_{3}.

(*a*_{1})^{4} – 2(*a*_{1})^{2 }= (*a*_{1})^{2}

Subtract (*a*_{1})^{2} from both sides.

(*a*_{1})^{4} – 3(*a*_{1})^{2 }= 0

Factor out (*a*_{1})^{2 }from both terms.

(*a*_{1})^{2} ((*a*_{1})^{2} – 3) = 0

This means that either (*a*_{1})^{2 }= 0, or (*a*_{1})^{2} – 3 = 0.

If (*a*_{1})^{2 }= 0, then *a*_{1} must be 0. However, we are told that all the terms of the sequence are positive. Therefore, the first term can't be 0.

Next, let's solve (*a*_{1})^{2} – 3 = 0.

Add 3 to both sides.

(*a*_{1})^{2 }= 3

Take the square root of both sides.

*a*_{1} = ±√3

However, since all the terms are positive, the only possible value for *a*_{1} is √3.

Now, that we know that *a*_{1} = √3, we can find *a*_{2}, *a*_{3}, and *a*_{4}.

*a*_{2} = (*a*_{1})^{2} – 1 = (√3)^{2} – 1 = 3 – 1 = 2

*a*_{3} = (*a*_{2})^{2} – 1 = 2^{2 }– 1 = 4 – 1 = 3

*a*_{4} = (*a*_{3})^{2} – 1 = 3^{2} – 1 = 9 – 1 = 8

The question ultimately asks for the product of the *a*_{2}, *a*_{3}, and *a*_{4}, which would be equal to 2(3)(8), or 48.

The answer is 48.

### Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

In the given sequence, the first term is 3 and each term after is one less than three times the previous term.

What is the sixth term in the sequence?

**Possible Answers:**

**Correct answer:**

The fourth term is: 3(23) – 1 = 69 – 1 = 68.

The fifth term is: 3(68) – 1 = 204 – 1 = 203.

The sixth term is: 3(203) – 1 = 609 – 1 = 608.

### Example Question #2 : Nth Term Of An Arithmetic Sequence

Consider the following sequence of numbers:

What will be the 8^{th} term in the sequence?

**Possible Answers:**

49

60

56

58

51

**Correct answer:**

51

Each number in the sequence in 7 more than the number preceding it.

The equation for the terms in an arithmetic sequence is a_{n} = a_{1} + d(n-1), where d is the difference.

The formula for the terms in this sequence is therefore a_{n} = 2 + 7(n-1).

Plug in 8 for n to find the 8^{th} term:

a_{8} = 2 + 7(8-1) = 51

### Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

The second and fourth terms of an arithmetic sequence are 9 and 18, respectively. What is its first term?

**Possible Answers:**

**Correct answer:**

The difference between the second and fourth terms of an arithmetic sequence is twice the common difference - or, equivalently, the common difference is half the difference between the second and fourth terms.

The common difference can be subtracted from the second term to obtain the first term:

### Example Question #171 : Integers

A sequence of numbers is represented by the equation , where represents the th term in the sequence. Which of the following equals the term in the sequence?

**Possible Answers:**

**Correct answer:**

Take the equation that represents the th term in the sequence and plug in the value of 9 for :

The value of the 9th term is 747.

### Example Question #1 : How To Find The Nth Term Of An Arithmetic Sequence

You are given a sequence with the same difference between consecutive terms. We know it starts at and its 3rd term is . Find its 10th term.

**Possible Answers:**

**Correct answer:**

From the given information, we know , which means each consecutive difference is 3.

### Example Question #1 : How To Find The Next Term In An Arithmetic Sequence

**Possible Answers:**

**Correct answer:**

Each term in the sequence is one less than twice the previous term.

So,

### Example Question #1 : How To Find The Next Term In An Arithmetic Sequence

What is the next number in the following series: 0, 3, 8, 15, 24 . . . ?

**Possible Answers:**

35

37

32

40

41

**Correct answer:**

35

The series is defined by n^{2} – 1 starting at n = 1. The sixth number in the series then equal to 6^{2} – 1 = 35.

### Example Question #1 : How To Find The Next Term In An Arithmetic Sequence

A sequence of numbers is as follows:

What is the sum of the first seven numbers in the sequence?

**Possible Answers:**

248

1529

621

719

490

**Correct answer:**

621

The pattern of the sequence is (x+1) * 2.

We have the first 5 terms, so we need terms 6 and 7:

(78+1) * 2 = 158

(158+1) * 2 = 318

3 + 8 + 18 +38 + 78 + 158 + 318 = 621

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