First determine the type of the sequence
A sequence is an enumerated collection of objects in which repetitions are allowed and order matters.
Types of Sequences
1. Linear/Arithmetic sequences
Sequences where the difference between each term is the same. Terms add (or subtract) the same number each time.
Example
The number added or subtracted is called common difference. To find the common difference, substrate the ( n
1)th term from the nth term. The common difference is supposed to be the same for all substations if it's a linear sequence. If it's not the same, then the sequence is not a linear sequence.
To find the
nth term of a linear/arithmetic sequence, the formula below is used:
an=a1+(n−1)dwhere
a1= first term
n= the position you are finding
d= the common difference
(an−a(n−1)) For example to find the
5th term for the sequence,
3,5,7,9,...,... we can apply the above formula.
n=5a5=a1+(5−1)dd=5−3=7−5=9−7=2a1= first term =3Hence the
5th term
=3+(5−1)×2=3+4×2=11Let's determine if the given sequence is linear or not.
2,5,10,17,......,......Common Difference
=5−2=3 or
10−5=5 or
17−10=7, since the common differences are not the same, the sequence can't be a linear sequence.
2. Geometric progressions
Geometric sequences / progressions are similar to linear sequences, but instead of adding/subtracting the same number each time, you multiply (or divide) by the same number each time.
Example
The term multiplied or divided is called common ratio. The common ratio is calculated by dividing the
nth term by the
(n−1 )th term.
Thus for the above example
2,4,8,16, the common ratio can be calculated as
4∕2=8∕4=16∕8=2The common ratio for each consecutive terms must be the same. If not, then the sequence is not geometric.
Let's determine if the sequence in the question is geometric
3. Fibonacci sequences
Fibonacci sequences are sequences where each term is the sum of the previous two terms. In a Fibonacci sequence, the first two numbers must be chosen before starting the sequence. The most common sequence begins with 0 and 1 . This specific sequence of numbers generated by 0 and 1 are known as the "Fibonacci numbers".
Examples
Let's determine if the sequence in the question is a Fibonacci sequence
2,5,10,17,......,......The
3rd term is supposed to be
2+5=7 but its 10 , hence the sequence is not a Fibonacci sequence.
4. Quadratic sequences
Quadratics are expressions in which the highest power of
x is an
x2 term. The general term of a quadratic sequence is
ax2+bx+c. For example,
2x2+3x+2 is a quadratic because the highest power of
x is
x2. However,
x3−2x2 is not a quadratic, because although it contains an
x2 term, there is a higher power of
x (the
x3 ).
In quadratic sequences, the differences between the terms are not the same, however the difference of the differences are the same. This is the only way you can identify them.
Take the examples below. You will see that the differences between the terms (in red) are different.
However the differences of the red terms (in green) are the same.
Let's check if the sequence in the question is quadratic or not.
2,5,10,17,......,......251017+3+5+7 +2 +2 The second difference (the one in green color) is 2.
Hence
2a=2, solving for
a,a=2∕2=1 equation (1)
3a+b=U2−U1U1 (the first term)
=2U2 (the second term)
=5, hence the difference
(U2−U1)=5−2=3Hence
3a+b=3, substitute
a=1 from equation 1 in this equation.
3(1)+b=3,b=3−3=0, hence b=0Since we know
a and
b, we can find
ca+b+c=U1 U1 (the first term)
=2Hence
1+0+c=2,c=2−1=1, hence
c=1Un=an2+bn+cHence
Un=(1)n2+(0)n+11×n2=n2,0×n=0, hence
Un=n2+1 Now we can find the
5th and
6th terms
U5=52+1=25+1=26U6=62+1=36+1=37Hence the
5th and
6th terms are 26 and 37 respectively.
You can verify the already given terms with the obtained formula to prove the formula is correct
| n (Term) | n2+1 | Result |
|---|
| 1 | 12+1=1+1 | 2 |
| 2 | 22+1=4+1 | 5 |
| 3 | 32+1=9+1 | 10 |
| 4 | 42+1=16+1 | 17 |