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Arithmetic Sequence {51,52,53,54,...,151}










































Number of Terms

     • An ordered list of numbers is called a sequence.

     • The sum of the numbers in a sequence is called a series.

How many terms are there in the sequence shown below?

       {51,52,53,54,...,151}

Comments for Arithmetic Sequence {51,52,53,54,...,151}

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Sep 09, 2012
Arithmetic Sequence
by: Staff


Answer:


Your sequence {51,52,53,54,...,151} is called an arithmetic sequence because each term in the sequence is equal to the previous term plus a constant. In this case the constant = 1.

{51,52,53,54,...,151} = {51,52=51+1,53=52+1,54=53+1,...,151=150+1}

The constant (in this case, the constant of 1) is called common difference, and is usually represented by the variable “d”.

This sequence is a linear sequence. A plot of the terms looks like this:




Arithmetic Sequence {51,52,53,54,...,151}




You can calculate the nth term in an arithmetic sequence using the following formula:

an = a₁ + (n - 1) * (d)


an = a with a subscript of n (this is the nth term in the series)

a₁ = a with a subscript of 1 (this is the 1st term in the series)

n = number of terms

d = difference between consecutive terms (the common difference)


We are going to apply this formula to calculate the number of terms in the sequence.


For your problem:

The last term (n = unknown):

alast term = 151

The first term (n = 1):

afirst term = a₁ = 51

n = number of terms = unknown

d = difference between consecutive terms (the common difference) = 1


an = a₁ + (n - 1) * (d)

alast term = afirst term + (n - 1) * (d)

alast term = a₁ + (n - 1) * (d)

substitute the value 151 for alast term

substitute the value 51 for afirst term, or a₁

151 = 51 + (n - 1) * (1)


Now it is only a matter of solving for “n”.

151 = 51 + (n - 1) * (1)

Multiply (n - 1) * (1) to eliminate the parentheses

151 = 51 + (n - 1)

151 = 51 + n - 1


Add 1 to each side of the equation

151 + 1 = 51 + n - 1 + 1

152 = 51 + n + 0

152 = 51 + n


Subtract 51 from each side of the equation

152 - 51 = 51 + n - 51

152 - 51 = 51 - 51 + n

152 - 51 = 0 + n

152 - 51 = n

101 = n

n = 101 terms


Final Answer: n = 101 terms


-------------------------------------------------

Check the answer

If n = 101 you can calculate the last term using the formula:


alast term = a₁ + (n - 1) * (d)


a₁ = 51

n = 101

d = 1


alast term = 51 + (101 - 1) * (1)

alast term = 51 + (101 - 1)

alast term = 51 + (100)

alast term = 51 + 100


alast term = 151, OK (this is the last term given in the problem statement)




Thanks for writing.

Staff
www.solving-math-problems.com


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