-11, -9, -7, -5, -3 What is the next number? What is the 200th term in this sequence? We see that Term 1 is -11, Term 2 is -9, so we do a point slope equation of (1,-11)(2,-9) to get the [URL='https://www.mathcelebrity.com/search.php?q=%281%2C-11%29%282%2C-9%29']nth term of the formula[/URL] f(n) = 2n - 13 The next number is the 6th term: f(6) = 2(6) - 13 f(6) = 12 - 13 f(6) = [B]-1 [/B] The 200th term is: f(200) = 2(200) - 13 f(200) = 400 - 13 f(200) = [B]387[/B]

1, 1/2, 1/3, 1/4, 1/5 What is the next number? What is the 89th term of the sequence? Formula for nth term is 1/n Next number is n = 5, so we have [B]1/5[/B] With n = 89, we have [B]1/89[/B]

1, 1/2, 1/4, 1/8, 1/16 The next number in the sequence is 1/32. What is the function machine you would use to find the nth term of this sequence? Hint: look at the denominators We notice that 1/2^0 = 1/1 = 1 1/2^1 = 1/2 1/2^2 = 1/4 1/2^3 = 1/8 1/2^4 = 1/32 So we write our explicit formula for term n: f(n) = [B]1/2^(n - 1)[/B]

1, 8, 27, 64 What is the 10th term? We see the following pattern: 1^3 = 1 2^3 = 8 3^3 = 27 4^3 = 64 We build our sequence function using this pattern: f(n) = n^3 With n = 10, we have: f(10) = 10^3 f(10) = [B]1,000[/B]

3, 6, 12, 24, 48 What is the function machine for this sequence? We see the following pattern: 3 * 2^0 = 3 3 * 2^1 = 6 3 * 2^2 = 12 3 * 2^3 = 24 3 * 2^4 = 48 Our function machine for term n is: [B]f(n) = 3 * 2^(n - 1)[/B]

31,29,24,22,17 what comes next We see that each sequence term alternates between subtracting 2 and subtracting 5. Since the last term, 17, was found by subtracting 5, our next term is found by subtracting 2 from 17: 17 - 2 = [B]15[/B]

7 and 105 are successive terms in a geometric sequence. what is the term following 105? Geometric sequences are set up such that the next term in the sequence equals the prior term multiplied by a constant. Therefore, we express the relationship in the following equation: 7k = 105 where k is the constant [URL='https://www.mathcelebrity.com/1unk.php?num=7k%3D105&pl=Solve']Type this equation into our search engine[/URL] and we get: k = 15 The next term in the geometric sequence after 105 is found as follows: 105*15 = [B]1,575[/B]

7, 10, 15, 22 What is the next number in the sequence? What is the 500th term? We see that: 1^2 + 6 = 7 2^2 + 6 = 10 3^3 + 6 = 15 4^2 + 6 = 22 We build our function as f(n) = n^2 + 6 Next term in the sequence is f(5) f(5) = 5^2 + 6 f(5) = 25 + 6 f(5) = [B]31 [/B] Calculate the 500th term: f(500) = 500^2 + 6 f(500) = 250,000 + 6 f(500) = [B]250,006[/B]

8,11,14,17,20 What is the next number? What is the 150th term? We're adding by 3 to the last number in the sequence, so we have the next number as: 20 + 3 = [B]23 [/B] For the nth term, we have a formula of this: 3n + 5 3(1) + 5 = 8 3(2) + 5 = 11 3(3) + 5 = 14 With n = 150, we have: 3(150) + 5 = 450 + 5 = [B]455[/B]

9, 3, 1, 1/3, 1/9 What is the next number in this sequence? What is the function machine for this sequence? We see the following pattern in this sequence: 9 = 9/3^0 3 = 9/3^1 1 = 9/3^2 1/3 = 9/3^3 1/9 = 9/3^4 Our function machine formula is: [B]f(n) = 9/3^(n - 1) [/B] Next term is the 6th term: f(6) = 9/3^(6 - 1) f(6) = 9/3^5 f(6) = 9/243 f(6) = [B]1/27[/B]

A car’s purchase price is $24,000. At the end of each year, the value of the car is three-quarters of the value at the beginning of the year. Write the first four terms of the sequence of the car’s value at the end of each year. three-quarters means 3/4 or 0.75. So we have the following function P(y) where y is the number of years since purchase price: P(y) = 24000 * 0.75^y First four terms: P(1) = 24000 * 0.75 = [B]18000[/B] P(2) = 18000 * 0.75 = [B]13500[/B] P(3) = 13500 * 0.75 = [B]10125[/B] P(4) = 10125 * 0.75 = [B]7593.75[/B]

A new company is projecting its profits over a number of weeks. They predict that their profits each week can be modeled by a geometric sequence. Three weeks after they started, the company's projected profit is $10,985.00 Four weeks after they started, the company's projected profit is $14,280.50 Let Pn be the projected profit, in dollars, n weeks after the company started tracking their profits. a. What is the common ratio of the sequence? b. Calculate the initial value c. Construct a recurrence relation that can be used to model the value of Pn a. 14,280.50/10,985.00 = [B]1.3[/B] b. 3 weeks ago, the Initial value is 10,985/1.3^3 = [B]$5,000 c. Pn = 5000 * 1.3^n[/B]

Annie got a new video game. She scored 152 points on the first level, 170 points on the second level, 188 points on the third level, and 206 points on the fourth level. What kind of sequence is this? This is an [URL='https://www.mathcelebrity.com/sequenceag.php?num=152%2C170%2C188%2C206&n=10&pl=Calculate+Series&a1=&d=']arithmetic series as seen on our calculator[/URL]:

Free Arithmetic and Geometric and Harmonic Sequences Calculator - This will take an arithmetic series or geometric series or harmonic series, and an optional amount (n), and determine the following information about the sequence
1) Explicit Formula
2) The remaining terms of the sequence up to (n)
3) The sum of the first (n) terms of the sequence Also known as arithmetic sequence, geometric sequence, and harmonic sequence

Free Coin Toss Probability Calculator - This calculator determines the following coin toss probability scenarios
* Coin Toss Sequence such as HTHHT
* Probability of x heads and y tails
* Probability of at least x heads in y coin tosses
* Probability of at least x tails in y coin tosses
* Probability of no more than x heads in y coin tosses
* Probability of no more than x tails in y coin tosses
* (n) Coin Tosses with a list of scenario results displayed
* Monte Carlo coin toss simulation

Free Fibonacci Sequence Calculator - Generates a list of the first 100 Fibonacci numbers. Also shows how to generate the nth Fibonacci number using Binet's Formula

Find the explicit formula of the sequence 3,12,48 We [URL='https://www.mathcelebrity.com/sequenceag.php?num=3,12,48&n=10&pl=Calculate+Series']type in 3,12,48 into our search engine[/URL]. Choose series, and we get: [B]a(n) = 3 * 4^(n - 1)[/B]

Look at this sequence: 53, 53, 40, 40, 27, 27, ... What number should come next? This looks like a sequence where we subtract 13 and then 0, 13 and then 0 from the prior number. Since the last group of 27 repeated, our next number is found by subtracting 13: 27 - 13 = [B]14[/B]

Oakdale School is sponsoring a canned food drive. In the first week of the drive, the students collected 638 cans. They collected 698 cans in the second week and 758 cans in the third week. If the students continue to collect cans at this rate, in which week will they collect more than 1,000 cans? We have an arithmetic sequence where each successive term increases by 50. [URL='https://www.mathcelebrity.com/sequenceag.php?num=638%2C698%2C758&n=10&pl=Calculate+Series&a1=5&d=3']Using our sequence calculator[/URL], we find that week #8 is when the students cross 1,000 cans.

Free Sequences Calculator - Given a function a(n) and a count of sequential terms you want to expand (n), this calcuator will determine the first (n) terms of your sequence, {a₁, a₂, ..., a_n}

Suppose that Sn = 3 + 1/3 + 1/9 + ... + 1/3(n-2) a) Find S10 and S∞ b) If the common difference in an arithmetic sequence is twice the first term, show that Sn/Sm = n^2/m^2 a) Sum of the geometric sequence is a = 3 and r = 1/3 (a(1 - r)^n)/(1 - r) (3(1 - 1/3)^9)/(1 - 1/3) [B]S10 = 4.499771376[/B] For infinity, as n goes to infinity, the numerator goes to 1 so we have [B]S∞ = 3(1)/2/3 = 4.5[/B] b) Sum of an arithmetic sequence formula is below: n(a1 + an)/2 an = a1 + (n - 1)2a1 since d = 2a1 n(a1 + a1 + (n - 1)2a1)/2 (2a1n + n^2 - 2a1n)/2 n^2/2 For Sm m(a1 + am)/2 am = a1 + (m - 1)2a1 since d = 2a1 m(a1 + 1 + (m - 1)2a1)/2 (2a1m + m^2 - 2a1m)/2 m^2/2 Sn/Sm = n^2/m^2 (cancel the 2's) S10/S1 = 10^2/1^2 We know S₁ = 3 So we have 100(3)/1 [B]S10 = 300[/B]

The teacher is handing out note cards to her students. She gave 20 note cards to the first student, 30 note cards to the second student, 40 note cards to the third student, and 50 note cards to the fourth student. If this pattern continues, how many note cards will the teacher give to the fifth student? [LIST] [*]Student 1 has 20 [*]Student 2 has 30 [*]Student 3 has 40 [*]Student 4 has 50 [/LIST] The teacher adds 10 note cards to each student. Or, if we want to put in a sequence formula, we have: S(n) = 10 + 10n where n is the student number Simplified, we write this as: S(n) = 10(1 + n) The question asks for S(5) S(5) = 10(1 + 5) S(5) = 10(6) [B]S(5) = 60 [/B] If we wanted to simply add 10 and not use a sequence formula, we see that S(4) = 50. So S(5) = S(4) + 10 S(5) = 50 + 10 [B]S(5) = 60[/B]

Van needs to enter a formula into a spreadsheet to show the outputs of an arithmetic sequence that starts with 13 and continues to add seven to each output. For now, van needs to know what the 15th output will be. Complete the steps needed to determine the 15th term in sequence. Given a first term a1 of 13 and a change amount of 7, expand the series The explicit formula for an [I]arithmetic series[/I] is an = a1 + (n - 1)d d represents the common difference between each term, an - an - 1 Looking at all the terms, we see the common difference is 7, and we have a1 = 13 Therefore, our explicit formula is an = 13 + 7(n - 1) If n = 15, then we plug it into our explicit formula above: an = 13 + 7(n - 1) a(15) = 15 + 7(15 - 1) a(15) = 15 + 7 * 14 a(15) = 15 + 98 a(15) = [B]113[/B]

What is the 7th number in the following pattern: 3.2, 4.4, 5.6, 6.8, ... This is an arithmetic sequence with an increase amount of 1.2. Each term S(n) is found by adding 1.2 to the prior term. S(1) = 3.2 S(2) = 3.2 + 1.2 = 4.4 S(3) = 4.4 + 1.2 = 5.6 S(4) = 5.6 + 1.2 = 6.8 S(5) = 6.8 + 1.2 = 8.0 S(6) = 8.0 + 1.2 = 9.2 S(7) = 9.2 + 1.2 = [B]10.4[/B]

We can't add a common number of multiply a common number to solve this. But what happens if we start at the 3rd term and add the 2 prior terms? [LIST] [*]0 + 1= 1 [*]1 + 1 = 2 [*]1 + 2 = 3 [*]3 + 5 = 8 [*]5 + 8 = [B]13[/B] [/LIST] [B]This is known as the Fibonacci Sequence. [MEDIA=youtube]9SZ1TJGwyLw[/MEDIA][/B]