The sequence of numbers a1, a2, a3, ..., an,... is defined by for each integer n ? 1. What is the sum of the first 20 terms of this sequence?
(1+1/2)-1/20, (1+1/2)-(1/21+1/22), 1-(1/20+1/22), 1-1/22, 1/20-1/22
5 Explanations
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Radwa Fahmy
Why by using the formula
Sn= n (a1+an) /2 (the answer is 443/66)
We don't get the same answer (value)?!
Good question! The problem with using that formula is that it is only for arithmetic sequences. Keep in mind that arithmetic sequences are evenly spaced, and have a common difference. For example, the sequence 5, 10, 15, 20, 25 is an arithmetic sequence, where the common difference is 5. That sequence is evenly spaced.
Arithmetic sequences can be written in the form:
an = a1 + (n-1)d
The sequence we are dealing with in this practice problem is not evenly spaced, and it cannot be written in the form above. Thus, it is not an arithmetic sequence, and the formula you mentioned won't work.
Why can't we use the idea that a sum of a sequence is the number of items in the last multiplied by the average of the first and last pair? I almost got to the correct answer but then got tripped up when I thought I had to divide by 2.
Hi Jessica,
No that should not be something we assume. With a sequence, we SHOULD write out the terms looking for a pattern (if we don't see the pattern immediately) and these kind of cancellations are things we should keep in mind/be aware of. But problems will always be a little different, so it's important to figure out each problem on its own. Hope that makes sense :)
The takeaway here (for me) is to match your answer with answer choices available. The answer choices are not simplified, so neither should yours as the test taker.
On a similar note, don't do anything that's going to take a ridiculous amount of time! :P
A huge trap is heading down a path that takes up a lot of time. One reason to practice a lot of questions—make the mistake now so you don't make the mistake on test day. :)
5 Explanations