series

At first glance, we see powers of 2 in the denominator of every term except the first one. But if we remember 2^0 = 1, we get our breakthrough. 1/2^0 = 1/1 = 1 Therefore, we stagger the powers of 2 by 1 less than the term we are on: a(n) = 1/2^(n - 1)

36, 34, 30, 28, 24 … It alternates like this: -2 -4 -2 -4 Next numbers should subtract 2 24 - 2 = 22

Map this out as a function with term number n and value 1, 0 2, 7 3, 14 4, 21 The values jump by 7, but they do so as the n - 1 term. We have the series formula S(n) = 7(n - 1) The problem asks for S(1000) S(1000) = 7(1000 - 1) S(1000) = 7(999) S(1000) = 6,993

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: f(n) = 9/3^(n - 1) Next term is the 6th term: f(6)...

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: f(n) = 3 * 2^(n - 1)

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...

1/2, 3, 5&1/2, 8......203 What term is the number 203? We see the following pattern: 1/2 = 2.5*1 - 2 3 = 2.5*2 - 2 5&1/2 = 2.5*3 - 2 8 = 2.5*4 - 2 We build our function f(n) = 2.5n - 2 Set 2.5n - 2 = 203 Using our equation solver, we get: n = 82

10, 1,000, 100,000, 10,000,000 What power of 10 is the 80th term? We see the following pattern 10^1 = 10 10^3 = 1000 10^5 = 100,000 10^7 = 10,000,000 f(n) = 10^(2n - 1) We build the 80th term: f(80) = 10^(2(80) - 1) f(80) = 10^(160 - 1) f(80) = 10^159

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) = 1,000

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) = 31 Calculate the 500th term: f(500) =...

1, 4, 9, 16, 25 What is the next number? What is the 50th term? We see that 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25 We build a formula for the nth term: f(n) = n^2 The next number means n = 6th term: f(6) = 6^2 = 36 The 50th term means n = 50: f(50) = 50^2 = 2500

5, 14, 23, 32, 41....1895 What term is the number 1895? Set up a point slope for the first 2 points: (1, 5)(2, 14) Using point slope formula, our series function is: f(n) = 9n - 4 To find what term 1895 is, we set 9n - 4 = 1895 and solve for n: 9n - 4 = 1895 Using our equation solver, we...

100, 75, 50, 25, 0, -25 What is the next number? What is the 100th term? Using point slope, we get (1, 100)(2, 75) Our series function becomes f(n) = -25n + 125 The next term is the 7th term: f(7) = -25(7) + 125 f(7) = -175 + 125 f(7) = -50 The 100th term is found by n = 100: f(100) =...

-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 nth term of the formula f(n) = 2n - 13 The next number is the 6th term: f(6) = 2(6) - 13 f(6) = 12 - 13...

3, 8, 13, 18, .... , 5008 What term is the number 5008? For term n, we have a pattern: f(n) = 5(n - 1) + 3 Set this equal to 5008 5(n - 1) + 3 = 5008 Using our equation solver, we get: n = 1002

1.25, 2, 2.75, 3.5 What is the 100th term? The formula of nth term is: f(n) = 0.75n + 0.5 So the 100th term is: f(100) = 0.75(100) + 0.5 f(100) = 75 + 0.5 f(100) = 75.5

2, 4, 6, 8....1000. What term is the number 1000? Formula for nth term is 2n If 2n = 1000, then dividing each side by 2, we see that: 2n/2 = 1000/2 n = 500

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 1/5 With n = 89, we have 1/89

0,7,14,21 What is the next number? What is the 1000th term? We're adding 7 to the last term, so we get a next term of: 21 + 7 = 28 For our nth term, we notice a pattern for the nth term of: 7n - 7 n = 1 --> 7(1) - 7 = 0 n = 2 --> 7(2) - 7 = 7 n = 3 --> 7(3) - 7 = 14 For n = 1000, we have...

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 = 23 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...