# 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}

 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: 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 = 151The first term (n = 1):afirst term = a₁ = 51n = number of terms = unknownd = 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 termsubstitute the value 51 for afirst term, or a₁151 = 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 parentheses151 = 51 + (n - 1) 151 = 51 + n - 1 Add 1 to each side of the equation151 + 1 = 51 + n - 1 + 1 152 = 51 + n + 0 152 = 51 + n Subtract 51 from each side of the equation152 - 51 = 51 + n - 51 152 - 51 = 51 - 51 + n 152 - 51 = 0 + n 152 - 51 = n 101 = n n = 101 termsFinal Answer: n = 101 terms -------------------------------------------------Check the answerIf n = 101 you can calculate the last term using the formula:alast term = a₁ + (n - 1) * (d)a₁ = 51n = 101d = 1alast 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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