rewrite each series using sigma notation
9) 1+4+9+16+25+36
and
10) 4+8+12+16+20
9) ∑(n^2), where n ranges from 1 to 6.
10) ∑(4n), where n ranges from 1 to 5.
To rewrite the given series using sigma notation:
9) The series 1+4+9+16+25+36 can be written as ∑(n^2), where n ranges from 1 to 6.
In other words, the sum of the squares of the first 6 positive integers.
10) The series 4+8+12+16+20 can be written as ∑(4n), where n ranges from 1 to 5.
In other words, the sum of the multiples of 4 from 1 to 5.
To rewrite each series using sigma notation, we need to understand the pattern and formula for the terms in each series.
For the first series: 1+4+9+16+25+36
We notice that each term is the square of the corresponding number in the series (1, 2, 3, 4, 5, 6). Therefore, the sum of the squares of the first n natural numbers can be expressed using sigma notation as:
∑(n^2)
For the second series: 4+8+12+16+20
We notice that each term is obtained by multiplying 4 (the common difference between consecutive terms) with the corresponding number in the series (1, 2, 3, 4, 5). Therefore, the sum of an arithmetic series with a common difference of 4 can be expressed using sigma notation as:
∑(4n)
Remember that the index n starts from 1, as in these series the first term corresponds to n = 1.
You want the general term for each one.
1+4+9+16+25+36
you can easily see that this is the square of the whole numbers, that is n^2
so [sigma] n^2 , from n = 1 to 6
(write n=1 below the "sigma" sign, and 6 on top of it)
for the 2nd, you have an AP
with a = 4, and d = 4
term(n) = a + (n-1)d = 4 + (n-1)(4)
= 4n
I will let you put it together