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avatar+992 

Hi friends,

 

just a question...If I look at the sums of each row in Pascal's triangle, I see a ratio of 2. Is it therefore save to say that a Geometric sequence is formed in the triangle?

 Mar 13, 2023
 #1
avatar
+1

The following are properties of Pascal's triangle:

 

1 - The sum of the nth row is 2^n. 
2 - The sum of rows 0 through n is 2^(n + 1) - 1. 
3 - The generating function of the nth row is (x + 1)^n. 

 

1 - Means the sum of, for example 5th row, ==2^5 ==32

2 - Means the sum of all rows, say from 0 to 5th row, ==2(5 + 1) - 1 ==2^6 - 1==63

3 - Means the generating function of the, say the 5th row, is: (x + 1)^5 ==1 x^5 + 5 x^4 + 10 x^3 + 10 x^2 + 5 x + 1 ==which means the coefficients of the expansion (as you can see) are: 1,  5, 10,  10,  5,  1 - which is the 5th row of Pascal's triangle.
 

 Mar 13, 2023
 #2
avatar+992 
+1

Thank you guest,

 

still no idea what all thatmeans, but thank you...do appreciate your time.

juriemagic  Mar 14, 2023
 #3
avatar+992 
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Could you perhaps indicate what the sum of the 10th row would be?

juriemagic  Mar 14, 2023
 #4
avatar+118459 
+1

 

 

The sum of the 0 row is 2^0 = 1

the sume of the nth row is  2^n

the sum of the 10th row is 2^10  = 1024

the sum of the 20th row is  2^20

etc

 

 Mar 14, 2023
edited by Melody  Mar 14, 2023
 #5
avatar+992 
+1

So Melody,

 

How do we go about finding what type of sequence is generated here?

juriemagic  Mar 14, 2023
 #6
avatar+118459 
+1

 it seems to be common pracice to accept that the first line is the 0 line, so it is easiest just to ignore that one

 

the sequence is    2,4,8,16,32

 

I can just see that this is   2^n    but if that is not immediately obvious to you, you can inspect it as a sequence.

 

since  4/2 = 8/4 = 16/8 = 32/16 = 2

 

this is a GP   

the first term is 2 and the common difference is 2

 

\(T_n=ar^{(n-1)}\\~\\ T_n=2*2^{(n-1)}\\~\\ T_n=2^{(n-1+1)}\\~\\ T_n=2^n\)

 Mar 15, 2023
 #7
avatar+992 
+1

Thank you Melody...you really are there for me every time!!!...Stay blessed

juriemagic  Mar 15, 2023

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