Answer:
The common difference is -1.
The last term (the 14th term) is -11.5.
Step-by-step explanation:
In an arithmetic sequence, the second term is 0.5 and the sum of the first 14 terms is -70.
We want to determine the: (a) common difference and (b) the last term.
We can write an explicit formula to represent the sequence. An arithmetic sequence can be modeled by the formula:
Where <em>a</em> is the initial term, <em>d </em>is the common difference, and <em>n</em> represents the <em>n</em>th term.
Since the second term is 0.5:
Simplify:
The sum of an arithmetic sequence is given by the formula:
Where <em>k</em> is the number of terms and <em>x_k</em> is the last term.
Since the sum of the first 14 terms is -70, <em>S</em> = -70 and <em>k </em>= 14:
Using our explicit formula, the last term is:
Substitute:
Simplifiy:
Rewrite the equation for the second term:
Substitute:
Simplify:
Solve for <em>d: </em>
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Hence, our common difference is -1.
Solve for <em>a</em>, the initial term:
So, our explicit formula is now:
So, the last term (which is 14) is:
