Sequences and Series - SSAT Upper Level Quantitative (Math)

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Question

Which of the following can be the sum of four consecutive positive integers?

Answer

Let , , , and be the four consecutive integers. Then their sum would be

In other words, if 6 were to be subtracted from their sum, the difference would be a multiple of 4. Therefore, we subtract 6 from each of the choices and see if any of the resulting differences are multiples of 4.

$178 -6 = 172 ; 172 \div4 = 43$

$213 -6 = 207 ; 207 \div4 = 51 ; R ; 3$

$420 -6 = 414 ; 414 \div4 = 103 : R : 2$

$143 -6 = 137 ; 137 \div4 = 34: R: 1$

Since this only happens in the case of 178, this is the only number of the four that can be a sum of four consecutive integers: 43, 44, 45, 46.

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Question

Three consecutive integers have sum . What is their product?

Answer

Let the middle integer of the three be . The three integers are therefore

, and they can be found using the equation

$\left (x-1 \right )+ x +\left ( x+1 \right ) = 427$

$x = 142\frac{1}{3}$

This contradicts the condition that the numbers are integers. Therefore, three integers satisfying the given conditions cannot exist.

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Question

Three consecutive odd integers have sum 537. What is the product of the least and greatest of the three?

Answer

Let the middle integer of the three be . The three integers are therefore

, and they can be found using the equation

$\left (x-2 \right )+ x +\left ( x+2 \right ) = 537$

The three integers are 177, 179, and 181, and the product of the least and greatest is

$177 \times 181 = 32,037$

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Question

Three consecutive even integers have sum 924. What is the product of the least and greatest of the three?

Answer

Let the middle integer of the three be . The three integers are therefore

, and they can be found using the equation

$\left (x-2 \right )+ x +\left ( x+2 \right ) = 924$

The three even integers are therefore 306, 308, and 310, and the product of the least and greatest of these is

$306 \times 310= 94,860$

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Question

Four consecutive integers have sum 3,350. What is the product of the middle two?

Answer

Call the least of the four integers . The four integers are therefore

,

and they can be found using the equation

$x + \left (x+1 \right )+\left (x+2 \right )+\left ( x+3 \right )= 3,350$

$x=3,344 \div 4 = 836$

The integers are 836, 837, 838, 839.

To get the correct response, multiply:

$837 \times 838 = 701,406$

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Question

Three consecutive integers have a sum of . What is their product?

Answer

Let the middle integer of the three be . The three integers are therefore

, and they can be found using the equation

$\left (x-1 \right )+ x +\left ( x+1 \right ) = 312$

The integers are therefore 103, 104, 105. The correct response is their product, which is

$103 \times 104 \times 105= 1,124,760$

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Question

What is the value of is this sequence?

Answer

This is a geometric sequence since the pattern of the sequence is through multiplication.

You have to multiple each value by to get the next one.

The value before is so .

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Question

The angle measures of a pentagon form an arithmetic sequence. The smallest angle measures $70^{\circ }$ . What does the largest angle measure?

Answer

The measures of the five angles of a pentagon add up to , based on the formula .

If the measures of the five angles form an arithmetic sequence, the angles have measures increasing by a some common difference, .

Given this pattern and the total sum, we can solve for the common difference.

$70+ \left( 70+d\right )+\left( 70+2d\right )+\left( 70+3d\right )+\left( 70+4d\right ) = 540$

The greatest of the angle measures is given by .

$70+4d = 70 + (4 )(19) = 146^{\circ }$

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Question

Find the $8^{th}$ term of the arithmetic sequence:

Answer

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

Now, plug in the information to find the value of the 8th term.

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Question

Find the $16^{th}$ term of the following arithmetic sequence:

Answer

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

Now, plug in the information to find the value of the 16th term.

$y_{16}=100+(-6)(16-1)=100+(-6)(15)=100-90=10$

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Question

Find the $9^{th}$ term of the following arithmetic sequence:

Answer

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

Now, plug in the information to find the value of the 9th term.

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Question

Find the $12^{th}$ term of the following arithmetic sequence:

Answer

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

Now, plug in the information to find the value of the 12th term.

$y_{12}=-1+(12-1)(5)=-1+(11)(5)=-1+55=54$

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Question

Find the $7^{th}$ term for the following arithmetic sequence:

Answer

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

Now, plug in the information to find the value of the 7th term.

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Question

Find the $10^{th}$ term of the following arithmetic sequence:

Answer

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

For this sequence,

Now, plug in the information to find the value of the 10th term.

$y_{10}=-15+(8)(10-1)=-15+(8)(9)=-15+72=57$

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Question

Brandon is improving very quickly at math. He improves at a rate of points per test. If he scored a on his first test, on his second test, and on his third test, what score will he get on his $8^{th}$ test?

Answer

You should recognize this as an arithmetic sequence:

The question is asking you to find the 8th term in this particular sequence.

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

Now, plug in the information to find the value of the 8th term.

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Question

Julia gets better every time she plays basketball. In her first game, she scored points. In her second game, she scored points, and in her third game, she scored points. If she continues to improve her basketball skills at this same pace, how many points should she be scoring by her $12^{th}$ game?

Answer

You should recognize this as an arithmetic sequence:

The question is asking you to find the 12th term in this particular sequence.

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

Now, plug in the information to find the value of the 12th term.

$y_{12}=8+(12-1)(4)=8+(11)(4)=8+44=52$

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Question

Leon makes for the first hour of work, for his second hour of work, for his third hour of work, and so on. How much will he make for his $12^{th}$ hour of work?

Answer

You should recognize this as an arithmetic sequence:

The question is asking you to find the 12th term in this particular sequence.

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

Now, plug in the information to find the value of the 12th term.

$y_{12}=2+(12-1)(4)=2+(11)(4)=2+44=46$

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Question

Find the sum of the first terms of the following arithmetic sequence:

Answer

To find the sum of a certain number of terms in an arithmetic sequence, use the following formula:

$\text{Sum}=\frac{n(y_1+y_n)}{2}$

the number of the terms you have

the first term of the sequence

$y_n=n^{th}$ term of the sequence

To find the sum, we need to first find the 8th term of the sequence.

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

Now, plug in the information to find the value of the 8th term.

Now that we know the 8th term of the sequence, we can plug in that value into the equation for the sum to find what these first 8 terms add up to.

$\text{Sum}=\frac{8(-10+32)}{2}=\frac{(8)(22)}{2}=88$

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Question

Find the sum of the first terms of the following arithmetic sequence:

Answer

To find the sum of a certain number of terms in an arithmetic sequence, use the following formula:

$\text{Sum}=\frac{n(y_1+y_n)}{2}$

the number of the terms you have

the first term of the sequence

$y_n=n^{th}$ term of the sequence

To find the sum, we need to first find the 9th term of the sequence.

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

Now, plug in the information to find the value of the 9th term.

Now that we know the 9th term of the sequence, we can plug in that value into the equation for the sum to find what these first 9 terms add up to.

$\text{Sum}=\frac{9(15+(-17))}{2}=\frac{(9)(-2)}{2}=-9$

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Question

Hunter makes for the first hour of work, for his second hour of work, for his third hour of work, and so on. How much money will he make if he works for hours?

Answer

The question is basically asking you to find the sum of the first terms of the following arithmetic sequence:

To find the sum of a certain number of terms in an arithmetic sequence, use the following formula:

$\text{Sum}=\frac{n(y_1+y_n)}{2}$

the number of the terms you have

the first term of the sequence

$y_n=n^{th}$ term of the sequence

To find the sum, we need to first find the 11th term of the sequence.

To find any term in an arithmetic sequence, use the following formula:

is the term we want to find

is the first term of the sequence

is the number of the term we want to find

is the common difference

Using the information given from the question,

Now, plug in the information to find the value of the 11th term.

$y_{11}=4+(11-1)(2)=4+(10)(2)=4+20=24$

Now that we know the 11th term of the sequence, we can plug in that value into the equation for the sum to find what these first 11 terms add up to.

$\text{Sum}=\frac{11(4+24)}{2}=\frac{11(28)}{2}=154$

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