Pre-Calc Sequences and Series

Arithmetic

100

Write the first 5 terms of the arithmetic sequence:

a₁=4, and d=3

4, 7, 10, 13, 16

200

Find the sum from j=1 to j=10 of 2j-3.

80

300

Find the sum of the first 100 positive multiples of 5.

25,250

400

Find a formula for a_n the arithmetic sequence, where a₇ = 8 and a₁₃ = 6.

a_n = 10 - 1/3(n-1)

500

The starting salary for an accountant is $34,000 with a guaranteed salary increase of $2250 per year.  Find the salary during the fifth year and the total compensation through 5 years of employment.

$43,000 in the 5th year

$192,500 total compensation

Geometric

100

Find a formula for the nth term of the geometric sequence where a₁ = 100 and r = 1.05

a_n = 100(1.05)^n-1

200

Determine whether 1/3, -2/3, 4/3, -8/3, ...

is geometric and if so find the common ratio.

Geometric, r = -2

300

Write the first 5 terms of the geometric sequence where a₁ = 2 and a₃ = 12

2, 2root6, 12, 12root6, 72

400

Find the sum of the infinite geometric series where

a₁ = 4, r = 2/3

12

500

A machine that costs $120,000 will depreciate by 30% per year over the next 5 years.  What is the value of the machine after the 5 years?

$20,168.40

Induction

100

Find a formula for the sum of the first n terms of:

9, 13, 17, 21, ...

S_n = n(2n+7)

200

In the proof of 3 + 5 + 7 + ... + (2n+1) = n(n+2) what is the first step?

Show it's true for n=1,

3 = 1(1+2) -> 3 = 1(3) -> 3=3

300

In the proof of 3 + 5 + 7 + ... + (2n+1) = n(n+2) what is the second step?

Assume that S_k = k(k+2)

400

In the proof of 3 + 5 + 7 + ... + (2n+1) = n(n+2) what is the third step?

Showing that S_k+1 = (k+1)(k+3)

500

Prove 3 + 5 + 7 + ... + (2n+1) = n(n+2).

1. 3=1(1+2) true for n=1

2. assume S_k = k(k+2)

3. S_k+1 = S_k + (2(k+1)+1)

S_k+1 = k(k+2) + (2k+3)

S_k+1 = k² + 4k + 3

S_k+1 =(k+1)(k+3)

True for all positive integers n.

Binomial Expansion

100

Determine the value of ₈C₅.

56

200

Expand and write in simplified polynomial form:

(x+4)⁴

x⁴ + 16x³ + 96x² + 256x + 256

300

Expand and write in a+bi form:

(5+2i)⁴

41+840i

400

In the expansion of (x²+y)¹⁰ what is the coefficient of the x⁸y⁶ term?

210

500

Write the full expansion of (3x²+2y³)⁵

243x¹⁰ + 810x⁸y³ + 1080x⁶y⁶ + 720x⁴y⁹ + 240x²y¹² + 32y¹⁵

Mixed Review

100

1 + 5

6

200

Simplify 45/75

3/5

300

How many siblings does Balaji have?

4

400

f(x)=3(1.1)^x  find f(7.5) rounded to the nearest thousandth.

6.131

500

Write the equation of a line passing through the point (4, 5) and (8, 17).

y = 3x-7