Sequences and Series -- Arithmetic and Geometric Sequences

Sequences are fun to learn and not really that difficult. 
There are many similarities between arithmetic and geometric sequences, so 
learn both together. 

Enjoy !!!!! 

From Mthcounts Mini: Sequences and Series

Easier concepts:

Sequences

Arithmetic sequence/determine the nth term

Arithmetic and geometric sequences

Mathcounts strategies : review some sums 

Note : Don't just memorize, but really understand the concepts.

Harder concepts:

Sum and Average of An Evenly Space

Relationship between arithmetic sequences, mean and median

Sequences, series and patterns

Some Common Sums

This Week's Work : Week 4 -- for Inquisitive Young Mathleges

First of all, problem of the week from Evan, a 5th grader:
A man notices a sign in Shop-a-Lot that says: "All prices are marked 25% off today only!" He decides to buy a shirt that costs $65.12 before the discount. He then uses a $16 gift certificate and the clerk applies 12% sales tax. What is the final cost of the shirt after all the steps are applied? Express your answer to the nearest hundredth. 

This week, we'll learn two very common sequences : arithmetic and geometric sequences.
There are quite a few similarities between these two types and they are closely linked to ratio, proportion
so just watch the videos and play around/generate a few/ponder on those sequences. I don't expect you to learn them in just one week.

Notes from Regents Exam Prep: Arithmetic and Geometric Sequences and Series

From Mthcounts Mini:

Easier concepts:

Sequences

Arithmetic sequence/determine the nth term

Arithmetic and geometric sequences

Harder concepts:

Relationship between arithmetic sequences, mean and median

Sequences, series and patterns

From my blog : 
Some special arithmetic sequences and the easier way to find their sum

Write some notes of the most important features of the arithmetic sequence.
The best note will be posted here to share with other students.
Be creative !!

2013 Mathcounts State Prep: Harder State Questions

Check out Mathcounts here, the best competition math program for middle school students.
Download this year's Mathcounts handbook here.

2004 Mathcounts State Sprint #19: The points (x, y) represented in this table lie on a straight line. The point (13, q) lies on the same line. What is the value of p + q? Express your answer as a decimal to the nearest tenth. 

 

Solution: 

#19:  Look at the table and you'll see each time x + 2, y would -3. 

-5 to -14 is (-9), three times (-3) so p = 2 + 3 x 2 = 8

p would = 13 when 2 + 5.5 * 2 = 13 so q = -5  +  (5.5) * (-3) = -21.5

p + q = -13.5

2004 Mathcounts State Sprint #24: The terms x, x + 2, x + 4, ..., x + 2n form an arithmetic sequence, with x an integer. If each term of the sequence is cubed, the sum of the cubes is - 1197. What is the value of n if n > 3?

Solution: 

The common difference in that arithmetic sequence is 2 and the sum of the cubes is -1197, which means that these numbers are all odd numbers. (cubes of odd numbers are odd and the sum of odd terms of odd numbers is odd. )

(-5)³ + (-7)³ + (-9)³ = -1197  However, n is larger than 3 (given) so the sequence will look like this:

 (-9)³+ (-7)³+ (-5)³+ (-3)³+ (-1)³ + (1)³ + (3)³ = -1197

 x = - 9 and x + 2n = 3, plug in and you get -9 + 2n = 3;  n = 6

2013 Mathcounts Basic Concept Review: Some Common Sums/numbers

These are some common sums that appear on Mathcounts often.

\(1+2+3+\ldots +n=\dfrac {n\left( n+1\right) } {2}\)

 \(2+4+6+\ldots .2n=n\left( n+1\right)\)

\(1+3+5+\ldots .\left( 2n-1\right) =n^{2}\)

The above are all arithmetic sequences.

The sum of any arithmetic sequence is average times terms (how many numbers).
Besides, the mean and median are the same in any arithmetic sequence.
Combining these knowledge, along with distributive rules some times
(case in point, sum of multiples of n, etc...) will expedite the calculation.

The "nth" triangular number is \(\dfrac {n\left( n+1\right) } {2}\)

The sum of the first n triangular numbers is \(\dfrac {n\left( n+1\right) \left( n+2\right) } {6}\).

 \(1^{2}+2^{2}+3^{2}+\ldots +n^{2}\) = \(\dfrac {n\left( n+1\right) \left( 2n+1\right) } {6}\)

\(1^{3}+2^{3}+3^{3}+\ldots +n^{3}=\)\(\left[ \dfrac {n\left( n+1\right) } {2}\right] ^{2}\)