a To begin, let's carefully consider the given terms in the sequence to determine whether it is a geometric or arithmetic sequence. If there is a common difference between terms, the sequence is arithmetic. If not, it is geometric. Let's highlight the value of the first term as well, as we need that to write the rule for either sequence type.
-30 +15 → - 15 +15 → 0 ...There is a common difference of +15, so this sequence is arithmetic. We can use this information to write an equation for the nth term of the sequence. Explicit equations for the nth term of an arithmetic sequence follow a certain format.
a_n= a_1+( n-1) d
In this form, a_1 is the first term in the sequence, d is the common difference, and n is the position of the desired term in the sequence. For our sequence, the common difference d= 15 and the first term a_1= -30. Let's write and simplify the equation.
b We want to find an equation for the following sequence. 9, 3, 1, ...
There doesn't appear to be a common difference between terms, so we know that this sequence is not algebraic. Let's see if we can find a common ratio between terms to determine whether it is a geometric sequence.
There is a common ratio of 13 between terms in the sequence, which means that the sequence is geometric. Let's use this ratio and the first term 9 to write the formula. Remember, the explicit rule of a geometric sequence follows a set pattern.
a_n= a_1 r^(n-1)
In this form, a_1 is the first term in the sequence, r is the common ratio, and n is the position of the desired term in the sequence. Let's substitute our information into the rule to write our equation for the sequence.
a_n=a_1r^(n-1) ⇔ a_n= 9( 1/3 )^(n-1)
