in the xy-plane, the graph of the function f is a line. if f(2)=7 and f(12)=1, what is the value of f(7)?
1 answer:
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The difference between each term is -5, i.e. we know one of our terms is -5n. Now we just need to adjust it to get u1 correct. At n=1, -5n would give us -5 therefore we need to add 6 to get to 1. This gives us our equation:
Now we can work out the sum easily (we can use an equation using the above but personally I prefer the other one as it's less prone to mistakes):
We know n is 12, and we know u1 is 1, and we also know the difference is -5.
Substituting this in:
Now we can work out the sum easily (we can use an equation using the above but personally I prefer the other one as it's less prone to mistakes):
We know n is 12, and we know u1 is 1, and we also know the difference is -5.
Substituting this in:
All three series converge, so the answer is D.
The common ratios for each sequence are (I) -1/9, (II) -1/10, and (III) -1/3.
Consider a geometric sequence with the first term <em>a</em> and common ratio |<em>r</em>| < 1. Then the <em>n</em>-th partial sum (the sum of the first <em>n</em> terms) of the sequence is
Multiply both sides by <em>r</em> :
Subtract the latter sum from the first, which eliminates all but the first and last terms:
Solve for :
Then as gets arbitrarily large, the term will converge to 0, leaving us with
So the given series converge to
(I) -243/(1 + 1/9) = -2187/10
(II) -1.1/(1 + 1/10) = -1
(III) 27/(1 + 1/3) = 18