Exercise 4 Page 439

Explicit equations for arithmetic sequences follow a specific format. a_n= a_1+( n-1) d In this form, a_1 is the first term in a given sequence, d is the common difference from one term to the next, and a_n is the nth term in the sequence. For this exercise, the first term is a_1= 13. To find the common difference d, we will subtract the first term from the second term. a_2-a_1=1/2-1/3=3/6-2/6=1/6 We can observe the other terms to confirm that this is actually the common difference. 1/3+ 16 ⟶1/2+ 16 ⟶2/3+ 16 ⟶5/6... By substituting d= 16 and a_1= 13 into the explicit equation and simplifying, we can find the formula for this sequence.

a_n=a_1+(n-1)d

SubstituteII

d= 1/6, a_1= 1/3

a_n= 1/3+(n-1)( 1/6)

Distr

Distribute 1/6

a_n=1/3+1/6n-1/6

▼

Subtract fractions

ExpandFrac

a/b=a * 2/b * 2

a_n=2/6+1/6n-1/6

SubFrac

Subtract fractions

a_n=1/6n+1/6

This equation can be used to find any term in the given sequence. To find a_(50), the 50th term in the sequence, we substitute 50 for n.

a_n=1/6n+1/6

Substitute

n= 50

a_(50)=1/6( 50)+1/6

MoveRightFacToNumOne

1/b* a = a/b

a_(50)=50/6+1/6

AddFrac

Add fractions

a_(50)=51/6

ReduceFrac

a/b=.a /3./.b /3.

a_(50)=17/2

The 50th term in the sequence is 172. Note that we can also write it as a mixed number.

a_(50)=17/2

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Write fraction as a mixed number

Rewrite

Rewrite 17 as 16+1

a_(50)=16+1/2

WriteSumFrac

Write as a sum of fractions

a_(50)=16/2+1/2

CalcQuot

Calculate quotient

a_(50)=8+1/2

AddTerms

Add terms

a_(50)=8 12