+0  
 
+4
1792
9
avatar+870 

Exercise 1

Let (un) be a geometric series, which common ratio r equals 5 and 1st term  u0=10.

 

  1. Calculate the value of the 10 first terms.
  2. Determinate the value of u50.
  3. Calculate the sum S=u0+u1+u2+u3+...+u49.

Exercise 2

Let (un) be a sequence defined by un+1=un×\(\frac{3}{2}\), and u0=2.

  1. Conjecture that (un) is geomatrical.
  2. Prove your conjecture.

Exercise 3

For each of the following sequences, conjecture if they are, or not, geometrical series;

- If they seem to be, prove it.

- If they aren't obviously, find a counterexample (just calculate the first 3 terms of the sequence).

  1. un=4n
  2. vn=n4
  3. wn=\({3^n}\over {5^{n+1}}\)
  4. yn=3+3n

Lesson on geometrical series: http://www.regentsprep.org/regents/math/algtrig/ATP2/GeoSeq.htm

 

Good luck !     wink

 Oct 9, 2015
edited by EinsteinJr  Oct 9, 2015
edited by EinsteinJr  Oct 9, 2015

Best Answer 

 #1
avatar+129852 
+5

Here's the first one :

 

1) 10, 50, 250, 1250, 6250, 31250, 156250, 781250, 3906250, 19531250

 

2) u50  = 10*5(50 - 1)  = 10*549 = 177635683940025046467781066894531250

 

3)  Sum of the first 49 terms =  u0 * [ 1 - rn ] / [ 1 - r ]   = 10 * [ 1 - 549] / [ 1 - 5]  =

 

44408920985006261616945266723632810

 

 

 

cool cool cool

 Oct 9, 2015
 #1
avatar+129852 
+5
Best Answer

Here's the first one :

 

1) 10, 50, 250, 1250, 6250, 31250, 156250, 781250, 3906250, 19531250

 

2) u50  = 10*5(50 - 1)  = 10*549 = 177635683940025046467781066894531250

 

3)  Sum of the first 49 terms =  u0 * [ 1 - rn ] / [ 1 - r ]   = 10 * [ 1 - 549] / [ 1 - 5]  =

 

44408920985006261616945266723632810

 

 

 

cool cool cool

CPhill Oct 9, 2015
 #2
avatar+870 
0

Thanks CPhill, your answer is exact.

You got

20/20

for the first exercise, plus a brownie:

You'll get some more if you find the answer for the two other exercises too. laugh

 Oct 9, 2015
 #3
avatar+870 
0

Thanks CPhill, your answer is exact.

You got

20/20

for the first exercise, plus a brownie:

You'll get some more if you find the answer for the two other exercises too. laugh

 Oct 9, 2015
 #4
avatar+870 
0

Thanks CPhill, your answer is exact.

You got

20/20

for the first exercise, plus a brownie:

You'll get some more if you find the answer for the two other exercises too. laugh

 Oct 9, 2015
 #5
avatar+870 
0

And excuse me, I don't understand why my reply was posted three times instead of just once...

 Oct 9, 2015
 #6
avatar+129852 
0

Here's my attempt at the second

1) It is geometric    the first few terms are  2, 3, 9/2, 27/4......   the generating "formula"  for any term is : 2*(3/2)(n - 1)

 

2)  We can prove this by  dividing the (n+ 1)th term by the nth term....we have :

 

[2(3/2) ((n - 1) + 1) ] / [ 2 (3/2)n-1 ] =    (3/2)n / (3/2)n-1  =  3/2   =  r = the common difference between terms

 

 

 

cool cool cool

 Oct 9, 2015
edited by CPhill  Oct 9, 2015
 #7
avatar+870 
0

Correct again.

20/20

for the second exercise.

Ah, and I almost forgot the brownie wink

 Oct 9, 2015
 #8
avatar+129852 
0

Here's my best guesses for the last one :

 

1) 4n   .....   not  geometric ......    the first few terms are :  4, 8, 12, 16.......

 

This is  an arithmetic series, instead   

 

2)  n4  .....not geometric  ......the first few terms are :   1, 16, 81, 256

 

Note that the geometric difference between the first and second terms = 16, but between the 2nd and 3rd terms is only 81 / 16 .....geometric series always have a constant difference between terms

 

3)   3n / 5 n + 1   let's examine the first few terms before making any speculation....we have.....

 

3/25, 9/125/, 27/ 625........

 

The ratio of the second term to the first = [9/ 125] /[ [3 / 25] =  3/5

The ratio of the 3rd to the second = 3/5

 

So....it appears to be geometric   ....proof....

 

[ 3 n + 1 / 5 n + 2 ] / [3n / 5 n + 1 ]  =    [ 3 n + 1 / 5 n + 2 ] * [ 5 n + 1 / 3n ]  =  3 / 5

 

4)  3 + 3n    .....here's the first few terms......

 

6, 9, 12,  15.........   this is obviously not geometric........the ratio between the 2nd term to the first = 3/2   ...but the ratio between the 3rd and 2nd = 12/9 = 4/3

 

 

cool cool cool

 Oct 9, 2015
 #9
avatar+870 
+5

This is exact. Your marks :

ExerciseMark
Ex.120/20
Ex.220/20
Ex.320/20
Average20/20
TOTAL60/60

You deserved a chocolate mufin for such a good report card !

P.S.: If you don't like chocolate or prefer another flavor, tell me, I'll find it. laugh

 Oct 9, 2015

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