Question 1117124: Find the following:
a. the 25th term in the arithmetic sequence given that a1 = 4 and a2 = 9
b. the sum of the first 60 terms of the arithmetic sequence 4, 7, 10, 13, ......
Found 2 solutions by greenestamps, Shin123:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
a.
The common difference d in the sequence is 9-4=5.
a(2) = a(1)+1(d) = a(1)+1(5) = 4+5 = 9 [the second term is the first term, plus the common difference 1 time]
a(3) = a(1)+2(d) = a(1)+2(5) = 4+10 = 14 [the third term is the first term, plus the common difference 2 times]
...
a(n) = a(1)+(n-1)(d) = a(1)+(n-1)(5) = 4+10 = 14 [the n-th term is the first term, plus the common difference (n-1) times]
Your textbook probably shows a formula something like "a(n) = a(1)+(n-1)(d)" for the n-th term in an arithmetic sequence. But don't just memorize the formula; look at what it says and see that it is just common sense based on the definition of an arithmetic sequence.
So the 25th term in your sequence is the first term, plus the common difference 24 times: 4 + 24(5) = 4+120 = 124.
Answer for part a: the 25th term is 124.
b.
The formula you probably have for the sum of the first n terms of an arithmetic sequence is probably an ugly one; I suggest you don't try to memorize it. Instead, think of finding the sum of the terms of an arithmetic sequence like this....
(1) the sum of the terms is the number of terms, multiplied by the average of all the terms; and
(2) because the terms of an arithmetic sequence are equally spaced, the average of all the terms is just the average of the first and last terms.
So to find the sum of the terms of an arithmetic sequence we only need to know the first and last terms and the number of terms.
4, 7, 10, 13, ... out to 60 terms.
The first term is 4; the common difference is 3; so the 60th term is the first term plus the common difference 59 times: 4+59(3) = 4+177 = 181.
The average of the first and last terms -- and therefore the average of all the terms, is (4+181)/2 = 185/2.
The sum of the first 60 terms is the number of terms, multiplied by the average of all the terms:
Answer for part b: the sum of the first 60 terms is 5550.
Answer by Shin123(626)  (Show Source):
|
|
|
