Ch. 3 Sequences, Sums, and Inductions

How do we find the rule for n, given it is arithmetic? a5 = 22 a12 = 64 an = ??

64-22 = 42 12-5 =7 42/7 =6

a3 = 9/4 a6 = 243/32 an = ???

9/4 * r^3 = 243/32 r^3 because a3 = 9/4 multiply both sides by 4/9 (9/4 * r^3)*4/9 = (243/32)*4/9 = r^3 = 27/8 take the cube root: r =3/2

Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, ... a. an = 128(1/2)^n-1 b. an = 128(2)^n-1 c. an = 1/2(128)^n-1 d. an = 128(1/2)^n

A. Explanation The equation for this geometric sequence where each term is found by dividing the previous by 2 (multiplying by 1 / 2), is the one that begins with the 128 and is then multiplied by 1 / 2 to the n - 1 power.

Find the sum of the arithmetic series 106 +100 + 94 + ... + 28 + 22. 960 1536 768 1,792 896

Explanation We know the first and last terms in this series, but to find the sum using the formula for an arithmetic series, we also need to know the number of terms (n). Using the formula for the last term (a_n) of an arithmetic sequence, a_n = a_0+nd, with a common difference (d) of -6 and a zeroth (a_0) term of 112 (6 greater than the first term), we get 22 = 112 - 6n, which simplifies to 15 = n. Using the sum formula and replacing n with 15, a_1 with 106, and a_n with 22, the sum equals 15/2 (106 + 22) = 960.

If a2 = 5 and a8 = 35, what is the value of a30? 158 145 245 150 185

Explanation Between the 5 and the 35, there are six terms. The difference between 35 and 5 is 30. As 30 divided by 6 is 5, so each step increases by 5 each time. Let's find the equation for this sequence: a(n) = dn + a(0) We know that d = 5. Going back from a2 to a0 results in two steps. The difference between a2 and a0 is 10. 5 - 10 = -5 a(n) = 5n - 5 Using this information, to get to the 30th term: a30 = 5(30) - 5 = 145.

What is the 83rd term of the sequence 91, 87, 83, 79, ... ( = a1, a2, a3, a4, ...)?

Explanation Since we don't have a0, we'll use the formula with an = d(n - 1) + a1. This gives: an = -4(n - 1) + 91. Plugging in 83: a83 = -4(83 - 1) + 91 = -4(82) + 91 = -328 + 91 = -237.

What is the rule for the nth term of the geometric sequence if the third term is 96 and the fifth term is 1,536? A. an = 24(2)^n-1 B. an = 6(4)^n-1 C. an = 4(6)^n D. an = 4(6)^n-1 E. an = 4(4) ^n-1

Explanation Using the given information, write two equations: a3 = 96 and a5 = 1,536. To get to the fifth term, multiply the third term by the common ratio, r, twice, giving another equation: 96*r2 = 1,536. Solving for this common ratio: r2 = 1,536 / 96 = 16. r = 4 Using this common ratio and dividing the third term two times by it, gives you 96 / 4 = 24 / 4 = 6 for the initial term. The rule for the nth term here is this: an = 6(4)n - 1

Find the sum of the arithmetic series 106 +100 + 94 + ... + 28 + 22. 1536 960 896 1,792 768

Explanation We know the first and last terms in this series, but to find the sum using the formula for an arithmetic series, we also need to know the number of terms (n). Using the formula for the last term (a_n) of an arithmetic sequence, a_n = a_0+nd, with a common difference (d) of -6 and a zeroth (a_0) term of 112 (6 greater than the first term), we get 22 = 112 - 6n, which simplifies to 15 = n. Using the sum formula and replacing n with 15, a_1 with 106, and a_n with 22, the sum equals 15/2 (106 + 22) = 960.

A brick wall contains 52 bricks in its bottom row and 49 bricks in the next row up from the bottom row. Each subsequent row contains 3 fewer bricks than the row immediately below it. If the wall contains 16 rows, how many bricks total make up the wall? 472 518 720 944 360

Explanation We know the number of terms (16) and the first term (52) in this series, but to find the sum using the formula n/2(a_1+a_n), we also need to know the final term (a_n). Using the formula for a the last term of an arithmetic sequence (a_n = a_0+nd) with a common difference (d) of 3 and a zeroth term of 55 (3 greater than the first term) gives a_n = 55-3(16), or a_n = 7. We can now use the sum formula: replacing n with 16, a_1 with 52, and a_n with 7, the sum equals 16/2 (52 + 7), which simplifies to 472.

The ______ sequence is a well-known sequence in which each term is equal to the sum of the previous two terms.

Fibonacci

sum = a1(1-r^n)/1-r

Finite Geometric Series Formula r is the common ratio and n is the number of terms in the series

Which formula is this given r represents ratio? an = a1 * r^(n-1)

Geometric Sequence

(a1/r-1) only if r is between 0 and 1

Infinite Geometric Series

1.) Do we want to prove something for a set of elements that is infinite? 2.) Would it be easy to prove the property for the first element in the set? 3.) If we were to assume the property was true for the first k elements, can we use that to show that it is also true for the (k + 1)st element?

Mathematical Induction

1.) Show the property is true for the first element in the set. This is called the base case. 2.) Assume the property is true for the first k terms and use this to show it is true for the (k + 1)st term. This is called the induction step.

Mathematical Induction Steps

An _____ sequence is a sequence that increases or decreases by a constant amount with each number.

arithmetic

Which formula is this given n is a given number, d is the difference and a0 is our starting point? an = dn + a0

arithmetic formula can also be written as an =d(n-1) + a1

An arithmetic sequence is a pattern of numbers that go up by the same amount each time, like 1, 2, 3, 4, 5, ... which means that an _____ _______ is what you get when you're asked to add up all the entries within this pattern, like 1+2+3+4+5+

arithmetic series

The ______ numbers sequence is a sequence of numbers formed by cubing integers. The first terms are 1, 8, 27, 64, 125, 216

cube

a_n = a_0+nd

formula for the last term of an arithmetic sequence where d is the common difference

A _______ sequence is a sequence in which each number is multiplied (or divided) by the same value to get the next number.

geometric

Arithmetic Series Sigma Notation

n =1 just means start at the first row or a1

A ______ is a string of things in order

sequence

The ____ numbers sequence is a sequence of numbers formed by squaring integers. The first terms are 1, 4, 9, 16, 25, 36

square

The ______ number sequence is a sequence of numbers formed by creating larger and larger equilateral triangles. The first terms of the triangular number sequence are 1, 3, 6, 10, 15

triangular

What's the difference between a sequence and a series?

when you take the terms of a sequence and start adding them up, you create a series. Also, both sequences and series can have either a finite or infinite number of terms.