Finding the nth term of a sequence

In summary, the nth term of the sequence is ##n^2+1##. This can be derived by finding the values of a and d which generate the given sequence and using them to replace a and d in the expression n/2(2a+(n-1)d). Another approach is to solve a system of equations using the sequence's differences, which leads to the quadratic formula for the nth term, ##n^2+1##.
  • #1

chwala

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Find the nth term of the sequence; ## 2, 5, 10, 17,26......## this is how i did it:

2, 5, 10, 17, 26 can form a difference of 3,5, 7,9...
whose nth term is 1 + 2n. now how can apply this to the original sequence to get the nth term?

 
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  • #2
chwala said:
Find the nth term of the sequence; ## 2, 5, 10, 17,26......## this is how i did it:

2, 5, 10, 17, 26 can form a difference of 3,5, 7,9...
whose nth term is 1 + 2n. now how can apply this to the original sequence to get the nth term?
Starting with the 2, you add numbers of the form 2r+1. Do you know how to write a summation expression for a sum of numbers of that form?
 
  • #3
haruspex said:
Starting with the 2, you add numbers of the form 2r+1. Do you know how to write a summation expression for a sum of numbers of that form?
thanks i understand the 2r+1 which is the 1+2n which i indicated, my interest here is how to get the nth term yes sum ##Sn= n/2(2a+(n-1)d)## how do i use this?
 
  • #4
chwala said:
thanks i understand the 2r+1 which is the 1+2n which i indicated, my interest here is how to get the nth term yes sum ##Sn= n/2(2a+(n-1)d)## how do i use this?
Find the values of a and d which generate the given sequence.
 
  • #5
haruspex said:
Find the values of a and d which generate the given sequence.
the nth term is ##n^2+1## now my problem is i know ##a=2## what about d? i have tried using ##d= 2n+1## and i am getting ##Sn= n/2(2n^2-2+3)##
now how do i move from here to the required ##n^2+1##?
 
  • #6
chwala said:
i have tried using d=2n+1
No, a and d are constants.
Write the first two terms in the sequence in terms of a and d. That gives you two equations, two unknowns.
 
  • #7
haruspex said:
No, a and d are constants.
Write the first two terms in the sequence in terms of a and d. That gives you two equations, two unknowns.
please assist me i still don't understand.....
 
  • #8
chwala said:
please assist me i still don't understand.....
You have an expression Sn=n/2(2a+(n−1)d) which you believe will generate the sequence given the right values of a and d. In particular, it will need to give the right values for S1 and S2. You know what S1 and S2 are, so that gives you two equations.
 
  • #9
haruspex said:
You have an expression Sn=n/2(2a+(n−1)d) which you believe will generate the sequence given the right values of a and d. In particular, it will need to give the right values for S1 and S2. You know what S1 and S2 are, so that gives you two equations.
for S1=n/2(2a+(n-1)d gives
2= 1/2(2.a+(1-1)d) gives 4=2a and S2=2+5=7, 7=2/2(2a+(2-1)d gives 7=2a+d sir, i still don't understand just make it clear for me i am conversant with the general knowledge on arithmetic and geometric...just give the needed equations where a=2 and d=3 how do you move from here to n^2+1?
 
  • #10
chwala said:
for S1=n/2(2a+(n-1)d gives
2= 1/2(2.a+(1-1)d) gives 4=2a and S2=2+5=7, 7=2/2(2a+(2-1)d gives 7=2a+d sir, i still don't understand just make it clear for me i am conversant with the general knowledge on arithmetic and geometric...just give the needed equations where a=2 and d=3 how do you move from here to n^2+1?

You were asked to give a formula for the nth term of the sequence; you have done that, so you have answered the question in full. Why do you think you need to "move from here to n^2+1"?
 
  • #11
chwala said:
a=2 and d=3
So use those to replace a and d in your expression n/2(2a+(n−1)d).
 
  • #12
haruspex said:
So use those to replace a and d in your expression n/2(2a+(n−1)d).
Thanks i got it......##(1+1)+3+5+7+9+.##...
= ##1+(1+3+5+7+9+....)## forms the series ##1+ n^2##
 
  • #13
This could also be solved by; system of equations. i.e

We have the sequence;
##[ 2,5,10,17,26,...)##
##[3,5,7,9,...)##
##[2,2,2,2,...)##

Since we went up to what i call second difference then the nth term will be quadratic therefore we shall be solving;

##2a=2, a=1##

##3a+b=3 ⇒b=0##

##a+b+c=2 ⇒ 2+0+c=2 ##

Therefore ## c=1##

thus;

##U_n= n^2+1##
 

1. What is the purpose of finding the nth term of a sequence?

The purpose of finding the nth term of a sequence is to be able to predict or calculate any term in the sequence without having to list out all the terms. This allows for easier and more efficient calculations and analysis of the sequence.

2. How do you find the nth term of a sequence?

To find the nth term of a sequence, you need to first identify the pattern or rule that governs the sequence. This can be done by looking at the difference between consecutive terms or by looking at the relationship between the terms. Once the pattern is identified, you can use it to create an algebraic expression that represents the nth term of the sequence.

3. Can there be more than one possible nth term for a sequence?

Yes, there can be more than one possible nth term for a sequence. This is because some sequences may have multiple patterns or rules that govern them. Therefore, it is important to carefully analyze the sequence and consider all possible patterns before determining the nth term.

4. Can the nth term of a sequence be a negative number?

Yes, the nth term of a sequence can be a negative number. This can occur when the pattern or rule of the sequence involves subtracting a number or when the sequence itself is a decreasing sequence.

5. Why is it important to find the nth term of a sequence?

Finding the nth term of a sequence is important because it allows for easier and more efficient calculations and analysis of the sequence. It also helps in predicting future terms in the sequence and understanding the underlying patterns and relationships within the sequence.

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