(a) What is the sum of the first 60 consecutive odd numbers?
Tell whether it illustrates inductive or deductive reasoning.
inductive reasoning deductive reasoning
(b) What is the sum of the first 1,200 consecutive odd numbers?
Tell whether it illustrates inductive or deductive reasoning.
inductive reasoning deductive reasoning
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(a) To find the sum of the first 60 consecutive odd numbers, we can use a formula for the sum of an arithmetic series. The formula is given by:
Sn = (n/2)(a + l)
where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term. In this case, since we are looking for the sum of the first 60 consecutive odd numbers, n = 60, a = 1, and l = 2n - 1.
Plugging in the values, we have:
Sn = (60/2)(1 + 2*60 - 1)
= 30(1 + 119)
= 30(120)
= 3600
Hence, the sum of the first 60 consecutive odd numbers is 3600.
Regarding the reasoning used to find the sum, this is an example of deductive reasoning. Deductive reasoning is based on established facts or general principles and uses logical steps to draw conclusions. In this case, we are using a specific formula for the sum of an arithmetic series to calculate the sum of the given numbers.
(b) As for the sum of the first 1,200 consecutive odd numbers, we can follow the same steps as in part (a) using the formula for the sum of an arithmetic series.
Sn = (n/2)(a + l)
where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term. Since we are looking for the sum of the first 1,200 consecutive odd numbers, n = 1,200, a = 1, and l = 2n - 1.
Plugging in the values, we have:
Sn = (1,200/2)(1 + 2*1,200 - 1)
= 600(1 + 2,399)
= 600(2,400)
= 1,440,000
Hence, the sum of the first 1,200 consecutive odd numbers is 1,440,000.
Similar to part (a), the reasoning used to find the sum is deductive reasoning.
a)
Sum(1 odd numbers) = 1
sum(first 2 odd numbers) = 1+3=4
sum(first 3 odd numbers) = 1+3+5 = 9
mmmhhh?
they look like perfect squares
sum(first 4 odd numbers) = 1+3+5+7 = 16
appears that sum(n odd numbers) = n^2
so Sum(first 60 odd numbers) = 60^2 = 3600
test it with an arithmetic sequence method
1+3+5+7 + ... 60 = ?
a = 1, d=2, n=60
sum(60) = (60/2)(2 + 59(2)) = 3600
ok, then!!!
According to your text or course notes, what kind of reasoning did I just use?
b) sum(1200) = 1200^2 = ....