Exercise 24 Page 319

The explicit formula of an arithmetic sequence combines the information provided by the two equations of the recursive form into a single equation. Recursive:& a_1= a_1, & a_n=a_(n-1)+ d; [0.8em] Explicit:& a_n= a_1+(n-1) dIn these formulas, d is the common difference and a_1 is the first term. Looking at the given recursive formula, we can identify the common difference d and the value of the first term a_1. a_1&=8; a_n=a_(n-1)- 12 & ⇕ a_1&= 8; a_n=a_(n-1)+( - 12) We can see that - 12 is the common difference and the first term is equal 8. Now we have enough information to form an explicit formula for this sequence.

a_n=a_1+(n-1)d

SubstituteII

a_1= 8, d= - 12

a_n= 8+(n-1)( - 12)

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Simplify right-hand side

Distr

Distribute - 12

a_n=8-12n+12

AddTerms

Add terms

a_n=- 12n+20