Exercise 10 Page 212

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 of the sequence, d is the common difference, and a_n is the nth term. In the given sequence, the first term is a_1= 1. Let's pay close attention to the pattern in order to determine the common difference d. 1+( - 1) ⟶0+( - 1) ⟶- 1+( - 1) ⟶- 2+( - 1) ⟶... The common difference is - 1. Therefore, by substituting a_1= 1 and d= - 1 into the explicit equation and simplifying, we can find the formula for this sequence.

a_n=a_1+(n-1)d

SubstituteII

d= -1, a_1= 1

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

Distr

Distribute -1

a_n=1-n+1

AddTerms

Add terms

a_n=- n+2

We found the equation for the nth term of the given arithmetic sequence. This equation can be used to find any term of the sequence. To find a_(25), the 25th term, we substitute 25 for n.

a_n=- n+2

Substitute

n= 25

a_(25)=- 25+2

AddTerms

Add terms

a_(25)=-23

The 25th term of the sequence is -23.