Did i do this problem correctly?

[abel muroi]

Find the first four partial sums and the nth partial sum of
the sequence a^n. ( ^ means exponent)

(n = the specific term)

a_n = 2/3^n (_ means sub)


2/3^1 = 2/3 (first term of sequence)

2/3^2 = 2/9 (second term of sequence)

2/3^3 = 2/27 (third term of sequence)

2/3^4 = 2/81 (fourth term of sequence)


s_n = 4/2 (2/3 + 2/81)

4/2 (4/3)

partial sum is 16/6

btw, how do i find the nth partial sum??

 

[abel muroi]

I thought s_n = n/2 (a_1 + a_n) was the general formula for finding the partial sum of a sequence.

(a_n is the value of the term)
(n is the number of the term)

What formula am I supposed to use?



What did I do wrong?

 

[HallsofIvy]

$\displaystyle \frac{a_1+ a_n}{n}$ is the sum of an arithmetic series only!

This is a geometric series. The first 4 terms are, as you say, 2/3, 2/9, 2/27, and 2/81.

The first four partial sums are 2/3, 2/3+ 2/9= 6/9+ 2/9= 8/9, 2/3+ 2/9+ 2/27= 18/27+ 6/27+ 2/27= 27/27= 26/27,
and 2/3+ 2/9+ 2/27+ 2/81= 54/81+ 18/81+ 6/81+ 2/81= 80/81. You should know the formula for the sum of a geometric sequence but even if you don't those examples should give you the idea.

 

[Steven G]

I thought s_n = n/2 (a_1 + a_n) was the general formula for finding the partial sum of a sequence.

(a_n is the value of the term)
(n is the number of the term)

What formula am I supposed to use?



What did I do wrong?

Suppose S = 1 + r +r^2 + r^3 + ... +r^n. Then rS= r +r^2 + r^3 + ... +r^n + r^(n+1).

Now subtract the two equations. The left side will be S-rS=(1-r)S. The right side will be.....

You want to know what S equals, but have what (1-r)S equals. So how do you solve this equation for S

This S will be your formula to calculate the first n terms (make sure you are starting with 1!)

 

[abel muroi]

Here's a simple example; these are the first 4 terms of a geometric series:
2 , 6 , 18 , 54

a = 1st term (2)
m = multiplyer (3)
n = number of terms (4)

Formula for sum:
Sum 1st n terms = a(m^n - 1) / (m - 1)
= 2(3^4 - 1) / (3 - 1)
= 80

Any idea why your teacher gave you that problem, but not the formula?

Another question:
2/3 + 4/7 = 14/21 + 12/21 = 26/21
Do you understand that...why the denominator is 21 ??

I understand why the denominator is 21, but I don't understand why the numerator is 14. I thought i was supposed to find the least common denominator between the fractions and then add the numerators..

 

[Steven G]

I understand why the denominator is 21, but I don't understand why the numerator is 14. I thought i was supposed to find the least common denominator between the fractions and then add the numerators..

You want to change the way 2/3 looks like. Most times when you want to change how a number (or an expression) looks, you multiply by 1. After all (2/3)*1=2/3 and 9*1 =9,....

There are many ways of writing 1. Like 3/3 or 9/9 or .4/.4 or (2/3)/(2/3). Generally if the numerator and denominator are equal to one another then the fraction equals 1.

Let's review multiplication of fractions. (2/5)*(3/7) =6/35. You just multiply the numerators and multiply the denominators.

We want to write 2/3 with a 21 in the denominator. So we need to multiply the 3 in the denominator by 7. If we just multiplied the 3 by 7, then we have 2/3 = 2/21, which is not correct (that is 2/3 is not 2/21)

We can only multiply 2/3 by 1 if we do not want to change its value. We write 2/3 = (2/3)*(7/7) = 14/21. Note that 7/7 = 1. In the end we are replacing 2/3 with something that equals 2/3, namely 14/21.

I hope this helps.

Attend all your classes and be certain that your instructor is qualified to teach arithmetic. If you question your instructors ability then you need to make a decision about what do this with this class.

 

[abel muroi]

Ah thank you I understand now. btw it's not that I have been skipping classes or that my teacher is useless, it's that I've ALWAYS had a problem with fractions (multiplying, dividing adding fractions)

I am currently taking pre calculus and before this class i took a algebra course a year prior. So i'm a little rusty on the concepts of fractions.