nth Term of an Arithmetic Sequence - SAT Math

The series is defined by n2 – 1 starting at n = 1. The sixth number in the series then equal to 62 – 1 = 35.

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

So,

The sequence provided is arithmetic. An arithmetic sequence has a common difference between each consecutive term. In this case, the difference is ; therefore, the next term is .

You can also use a formula to find the next term of an arithmetic sequence:

where the current term and the common difference.

First find the common difference of the sequence,

Thus there is a common difference of

between each term,

so follow that pattern for another terms, and the result is .

The pattern of this sequence is where represents the position of the number in the sequence.

for the first number in the sequence.

for the second number.

For the fourth term, . Therefore, .

The pattern of this sequence is where represents the place of each number in the order of the sequence.

Here are our givens:

, our first term.

, our second term.

, our third term.

This means that our fourth term will be:

.

Using the Product of Radicals principle, we can simplify the first two terms of the sequence as follows:

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

Add this to the second term to obtain the desired third term:

.

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

Setting :

The th term of an arithmetic sequence can be derived using the formula

Setting

Since no perfect square integer greater than 1 divides evenly into 5 or 10, both of the first two terms of the sequence are in simplest form.

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

Setting :

Add this to the second term to obtain the desired third term:

This is not among the given choices.

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

Setting :

The th term of an arithmetic sequence can be derived using the formula

Setting :

The decimal equivalent of this can be found by dividing 13 by 15 as follows:

The correct choice is .

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.

We start by multiplying 6 times 46, since the first 4 terms are already listed. We then add the product, 276, to the last listed term, 20. This gives us our answer of 296.

All answers in the sequence must end in a 5 or a 0.

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

The equation for the terms in an arithmetic sequence is an = a1 + d(n-1), where d is the difference.

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

Plug in 8 for n to find the 8th term:

a8 = 2 + 7(8-1) = 51

Let a1 represent the first term of the sequence and an represent the nth term.

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

a2 = a1 + 2

a3 = a1 + 2 + 2 = a1 + 2(2)

a4 = a1 + 3(2)

a5 = a1 + 4(2)

an = a1 + (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 = a1 + (a1 + 2) + (a1 + 2(2)) + (a1 + 3(2)) + (a1 + 4(2))

= 5a1 + 2 + 4 + 6 + 8

= 5a1 + 20

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

a5 - a1 = a1 + 4(2) – a1 = 8

Now, we set the two expressions equal and solve for a1.

5a1 + 20 = 8

Subtract 20 from both sides.

5a1 = –12

a1 = –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.

a10 = a1 + (10 – 1)(2)

a10 = –2.4 + 9(2)

= 15.6

The answer is 15.6 .

Let _a_1 be the first term in the sequence. We can use the fact that a n+1 = (an)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 = 22 – 1 = 4 – 1 = 3

_a_4 = (_a_3)2 – 1 = 32 – 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.

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

The difference between each term can be found through subtraction. For example the difference between the first and the second term can be found as follows:

One can check and see that this is the case for the other given numbers in the sequence as well.

In order to find the seventh term, expand the sequence by adding 14 to the last given number (4th number) and all of the following numbers until the 7th number in the sequence is reached.

This gives the sequence:

As seen above the seventh number in the sequence is 87 and the correct answer.

The purpose of this question is to understand the patterns of sequences.

First, an equation for the term in the sequence must be determined ().

This is true because

will create ,

will create ,

will create ,

will create .

Then, the eqution must be applied to find the specified term. For the tenth term, the expression must be evaluated, yielding 103.