Sequence and Series Review Problems

Sequence Review Problems Example 1: Find the explicit formula for the following arithmetic sequence: The explicit formula for arithmetic sequences is: You must replace the a1 and d 1, 6, 11, 16, … an = a1 + (n – 1)d

Sequence Review Problems First you must find the common difference between the numbers: 1, 6, 11, 16, … 5 5 d =

Sequence Review Problems Then find a1 a1 1, 6, 11, 16, …

Sequence Review Problems Plug them both into the equation: Distribute the d Combine like terms a1 = 1 an = 1 + (n – 1)5 d = 5 an = 1 + 5n – 5 an = 5n – 4

Practice 1 Put the following sequences into explicit form: 1, 3, 5, 7, … -2, 1, 4, 7, … 9, 6, 3, 0, … -67, -60, -53, …

Sequence Review Problems Example 2: Find the recursive formula for the following arithmetic sequence: The recursive formula for an arithmetic sequence is: You need to find the difference a1 = 3, a5 = 15 an = an-1 + d

Sequence Review Problems The difference formula: an = the second or last number in the sequence. It can be any number after the first number a1 represents the first number you’re given (if not a1, then use the earliest term)

Sequence Review Problems In this case we have a1 and a5, so a1 will be the first term and a5 will be the last a1 = 3, a5 = 15

Sequence Review Problems Plug d into the equation: an = an-1 + d an = an-1 + 3

Practice 2 Find the recursive formula for the following arithmetic sequences a1 = 2, a6 = 12 a1 = 17, a8 = -25 a3 = 5, a10 = 54 a7 = 23, a19 = -85

Sequence Review Problems Example 3: Find the explicit formula for the following geometric sequence: The explicit formula for a geometric sequence is: We must replace the a1 and r 1, 2, 4, 8, …

Sequence Review Problems r is the common ration determined by dividing the second from the first number: 1, 2, 4, 8, …

Sequence Review Problems And you know a1 is the first term in the series a1 1, 2, 4, 8, …

Sequence Review Problems Plug them both into the equation: DO NOT combine the numbers!!! In this case 1 times a number is just that numbers so you don’t need the 1 a1 = 1 r = 2

Practice 3 Find the explicit formula for the following geometric sequences 1, 3, 9, … 3, 12, 48, … 2, -10, 50, … 36, 18, 9, …

Sequence Review Problems Example 4: Find the recursive formula for the following geometric sequence: The recursive formula for a geometric sequence is 27, 9, 3, … an = r*an-1 times

Sequence Review Problems Find r by dividing the second term by the first 27, 9, 3, …

Sequence Review Problems Plug r into the equation an = r*an-1

Practice 4 Find the recursive formulas for the following geometric sequences 6, 12, 24, … 180, 60, 20, … 8, -32, 128, … -1, -2, -4, …

Series Review Problems Example 5: Find the sum of the arithmetic series This is an arithmetic series (any time you see the word SUM you’re using series) Since you have the first and last term, use the formula a1 = 3, a6 = 19

Series Review Problems The sum of the first n terms The number of terms you’re adding

Series Review Problems The last term you’re adding The first term you’re adding

Series Review Problems Since you have the first and the 6th term, there are a total of 6 terms you’re adding so a1 = 3, a6 = 19 n = 6 Plug everything into the equation

Series Review Problems * Simplify That is the sum of the first six numbers in the series

Series Review Problems Example 6: Find the sum of the following series: In this case we don’t have a1, so we’ll use a5 as the starting point a5 = 5, a30 = 78

Series Review Problems We need to find the number of terms we’re adding up We are starting with the 5th term, so we are leaving out terms 1, 2, 3, and 4 That means we are adding 30 – 4 terms So: n = 26

Series Review Problems In the formula, use a5 for the first number and a30 for the last Use your calculator a5 = 5, a30 = 78 n = 26

Practice 5 Find the sum of the following arithmetic series a1 = 2, a6 = 12 a1 = 17, a8 = -25 a3 = 5, a10 = 54 a7 = 23, a19 = -85

Series Review Problems Example 7: Find the 40th partial sum of the following arithmetic series: We aren’t given the last term in the series, so we can use the other arithmetic series formula: 1 + 4 + 7 + …

Series Review Problems 40th partial sum means add the first 40 terms so: We can also find d: n = 40 1 + 4 + 7 + … 3 3 d =

Series Review Problems Plug everything into your formula

Practice 6 Find the 20th partial sum of the series 1 + 5 + 9 + …

Series Review Problems Example 8: Find the sum of the geometric series: Use the geometric series formula: a1 = 2, r = 3, n = 6

Series Review Problems We have everything we need to plug in and solve the problem a1 = 2 r = 3 n = 6

Practice 7 Find the sum of the following geometric series a1 = 2, r = 4, n = 5 a1 = 3, r = -2, n = 10 a1 = 12, r = ½ , n = 6

Series Review Problems Put the following series in sigma notation: There are 5 terms so the bottom is n = 1 and the top is 5 1 + 4 + 7 + 11 + 15 a1 a5

Series Review Problems We now need an explicit formula to go in front The series is arithmetic since there is a common difference ? 1 + 4 + 7 + 10 + 13 3 3 d =

Series Review Problems Use the arithmetic explicit formula: an = a1 + (n – 1)d an = 1 + (n – 1)3 an = 1 + 3n – 3 an = 3n – 2

Series Review Problems Plug your formula into sigma 3n – 2

Practice 8 Put the following series into sigma notation 1 + 3 + 5 + 7 + 9 + 11 + 13 1 + 2 + 4 + 8 + 16 + 32 1 + -3 + -7 + -11

Series Review Problems Solve: n = 6 Start at the first term Add to the sixth term

Series Review Problems Since the formula is arithmetic (the n is not an exponent) we’ll use the formula: We need to find the first and last term (6th)

Series Review Problems To find the first term, plug 1 into the equation Now plug it all into your formula a1 =3(1) + 2 = 5 a6 =3(6) + 2 = 20 n = 6

Series Review Problems Solve

Series Review Problems Solve: Start at the 3rd term Add to the 8th term

Series Review Problems This is a geometric series since n is an exponent Use the formula: We need a1, r, and n

Series Review Problems Plug in 3 to find the first term we’re adding r is the number being raised to the power n is the number we are adding up: 8 – 2 = 6 (remember, we’re leaving out the first 2 numbers)

Series Review Problems Plug everything into the formula a3 = 4 r = 2 n = 6

Practice 9 Solve the following