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Hi, if there's anyone who could explain this to me, at least the first part, would be a great help! been going through some maths textbooks and none of them cover it.
Thanks!
(a) In an arithmetic progression, the 1st term is 13 and the 15th term is 111. Find
the common difference and the sum of the first 20 terms.
(b) Express the sum obtained in part (a) using the sigma notation.
Background info:Quote:
Originally Posted by bobchiba
if the terms of an arithmetic progression are a_1, a_2, a_3, ..., a_n, then the formula for an arithmetic progression is of the form:
http://upload.wikimedia.org/math/a/9...15ca84ae87.png
where a_n is the nth term of the sequence, a_1 is the first term, n is the current number of the term and d is the common difference.
example, we may write the fifth term as:
a_5 = a_1 + (5 - 1)d = a_1 + 4d
the sum of the first n terms is given by the formula:
http://upload.wikimedia.org/math/1/3...9449590a13.png
and we can express the sum of the first n terms in sigma notation by the following form:
http://upload.wikimedia.org/math/2/8...be6edb6f87.png
So the above is just stuff you have to memorize at this point. if you decide to go to higher mathematics you can find out what all this stuff really means. you may see Arithmetic progression for more info
now on to your questions, i have attached them in a diagram below.
EDIT: the last formula should be SUM{n=0 to n=19}13 + 7n, that is plug in the value of d
Thats awseome....thanks alot Jhevon, big help.
Hey jhevon...If you have the time, could you please go through how you express the sum in sigma notation, I don't quite follow the jump from expressing the sum to going into sigma.
Thanks.
Sure...Quote:
Originally Posted by bobchiba
remember what sigma notation means. when we say:
SUM{n = 0 to m} f(n), it means we do f(0) + f(1) + f(2) + ... + f(m)
so f(n) is a function that represents each term. we found that the function that represents each term was: a_n = 6 + 7n for n = 1,2,3,4...
so the first term is 6 + 7(1)
the second term is 6 + 7(2)
the third term is 6 + 7(3)
.
.
.
the 20th term is 6 + 7(20)
so what the sigma from n = 1 to 20 does, is add up each of these terms:
SUM{first 20 terms} a_n = [6 + 7(1)] + [6 + 7(2)] + ... + [6 + 7(20)]
so we can replace the number in the brackets by n, and say we want the sum from n=1 to n = 20. so the first term we plug in 1 for n then add when we plug 2 for n, then add when we plug in 3 for n and so on.
so the sigma notation will just be the formula for the nth term, and we go from n = 1 up how many terms we want to sum
the other one where i started at n = 0, i used the first formula we got, but it's the same principle.
so if we wanted say the sum of the first 56 terms of the arithmetic sequence a_n = 5 + 13n for n = 1,2,3,4...
that would be:
SUM{n = 1 to 56}5 + 13n