Key Questions
An arithmetic sequence is a sequence (list of numbers) that has a common difference (a positive or negative constant) between the consecutive terms.
Here are some examples of arithmetic sequences:
1.)7, 14, 21, 28 because Common difference is 7.
2.) 48, 45, 42, 39 because it has a common difference of - 3.The following are not examples of arithmetic sequences:
1.) 2,4,8,16 is not because the difference between first and second term is 2, but the difference between second and third term is 4, and the difference between third and fourth term is 8. No common difference so it is not an arithmetic sequence.
2.) 1, 4, 9, 16 is not because difference between first and second is 3, difference between second and third is 5, difference between third and fourth is 7. No common difference so it is not an arithmetic sequence.
3.) 2, 5, 7, 12 in not because difference between first and second is 3, difference between second and third is 2, difference between third and fourth is 5. No common difference so it is not an arithmetic sequence.
#a_n=a_1+(n-1)*d#
#color(white)("XXX")#where #a_1# is the first term and
#color(white)("XXXXXXX")d# is the difference between a term and its previous term.Examine the pattern:
#a_1#
#a_color(brown)(2)=a_1+d=color(green)(a_1+1d)#
#a_color(brown)(3)=a_2+d=a_1+d+d=color(green)(a_1+2d)#
#a_color(brown)(4)=a_3+d=a_1+2d+dcolor(green)(=a_1+3d)#
#a_color(brown)(5)=a_4+d=a_1+3d+d=color(green)(a_1+4d)#To find out the common difference in an AP you can perform the following simple step.
Subtract the first term of the AP from the second term of the AP.
d = #a_2 # - # a_1#
where d = common difference
# a_2# = any term other than first term
#a_1# = previous termFor example;
In the AP
3 , 9 , 15 , 21 , 27 , 33Taking #a_1# = 3
Taking #a_2# = 9#a_2# - #a_1# = 9 - 3 = 6
hence , common difference or d = 6
Thanks
I hope this helps
The even numbers, the odd numbers, etc
An arithmetic sequence is builded up adding a constant number (called difference) following this method
#a_1# is the first element of a arithmetic sequence, #a_2# will be by definition #a_2=a_1+d#, #a_3=a_2+d#, and so on
Example1:
2,4,6,8,10,12,....is an arithmetic sequence because there is a constant difference between two consecutive elements (in this case 2)
Example 2:
3,13,23,33,43,53,.... is an arithmetic sequence because there is a constant difference between two consecutive elements (in this case 10)
Example 3:
#1,-2,-5,-8,...# is another arithmetic sequence with difference #-3#
Hope this help
We think you wrote:
This solution deals with arithmetic sequences.
Find the common difference
Find the common difference by subtracting any term in the sequence from the term that comes after it.
The difference of the sequence is constant and equals the difference between two consecutive terms.
Find the sum
Calculate the sum of the sequence using the sum formula:
Plug in the terms.
Simplify the expression.
The sum of this sequence is .
This series corresponds to the following straight line
Find the explicit form
The formula for expressing arithmetic
sequences in their explicit form is:
Plug in the terms.
(this is the 1st term)
(this is the
common difference)
(this is the nth term)
(this is the term position)
The explicit form of this arithmetic sequence is:
Find the recursive form
The formula for expressing arithmetic sequences in their recursive form is:
Plug in the d
term.
(this is the common difference)
The recursive form of this arithmetic sequence is:
Find the nth element
Why learn this
When will the next bus arrive? How many people can fit inside a stadium? How much money will I earn this year? All these questions can be answered by learning how arithmetic sequences work. The progression of time, triangular patterns (bowling pins, for example), and increases or decreases in quantity can all be expressed as arithmetic sequences.
Terms and topics
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Algebra Examples
Popular Problems
Algebra
Identify the Sequence 3 , 10 , 17 , 24 , 31
, , , ,
Step 1
This is an arithmetic sequence since there is a common difference between each term. In this case, adding to the previous term in the sequence gives the next term. In other words, .
Arithmetic Sequence:
Step 2
This is the formula of an arithmetic sequence.
Step 3
Substitute in the values of and .
Step 4
Simplify each term.
Tap for more steps...
Apply the distributive property.
Multiply by .
Step 5
Subtract from .