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10,15,25,40, ....


how do u get the rule and what type of sequence is this

 May 29, 2021
 #1
avatar+26213 
+3

What is the rule of this sequence?
\(10,~15,~25,~40,~60,~\dots\)

how do u get the rule and what type of sequence is this

 

\(\begin{array}{|rcll|} \hline n & a_n & d_0 & d_1 \\ \hline 1 & 10 & \\ & & +5 \\ 2 & 15 & & +5 \\ & & +10 \\ 3 & 25 & & +5\\ & & +15 \\ 4 & 40 & & +5\\ & & +20\\ 5 & 60 & \\ \hline \end{array}\)

 

This is an Arithmetic Sequences Second order
\(\mathbf{an^2 + bn + c = a_n}\) with a, b, and c integers

 

\(\begin{array}{|lrcll|} \hline n=1: & a*1^2 + b*1 + c &=& 10 \\ n=2: & a*2^2 + b*2 + c &=& 15 \\ n=3: & a*3^2 + b*3 + c &=& 40 \\ \hline (1) & a+b+c &=& 10 \\ (2) & 4a+2b+c &=& 15\\ (3) & 9a +3b + c &=& 25 \\ \Rightarrow & a&=& \dfrac{5}{2} \\ \Rightarrow & b&=& -\dfrac{5}{2} \\ \Rightarrow & c&=& 10 \\ \hline & a_n &=& \dfrac{5}{2}*n^2 -\dfrac{5}{2}*n + 10 \\ \hline \end{array}\)

 

laugh

 May 29, 2021
 #2
avatar+2193 
+1

That's cool. 

I saw the pattern, but had no idea how to write it into a function. 

 

=^._.^=

catmg  May 29, 2021
 #3
avatar+34315 
+2

Question:    Can you just write this as a recursive sequence

    a1 =10   ,   an = an-1   + 5 ( n-1)            ?????

 

Thanx !    ~ EP

 May 29, 2021
 #4
avatar+121054 
+2

I  think  you  are correct, EP......excellent spot   !!!

 

The  series is

 

10,  15, 25, 40 , 60     .......

 

Heureka's  is  explicit   ( we  can find any term)

 

Yours  is  recursive   ( good  if  we  know  the nth term )

 

cool cool cool

CPhill  May 29, 2021
edited by CPhill  May 29, 2021
edited by CPhill  May 29, 2021

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