a) What is the sum of the first 70 consecutive odd numbers?
Tell whether it illustrates inductive or
deductive reasoning.
inductive reasoning deductive reasoning
What is the sum of the first 1,400 consecutive odd numbers?
Tell whether it illustrates
inductive or deductive reasoning.
inductive reasoning deductive reasoning
To find the sum of consecutive odd numbers, you can use the formula for the sum of an arithmetic series. The formula is given by:
Sum = n/2 * (first term + last term)
Where n is the number of terms in the series, the first term is the starting odd number, and the last term is the ending odd number.
a) For the first question, to find the sum of the first 70 consecutive odd numbers, you need to identify the first term and the last term.
The first term in a sequence of consecutive odd numbers is 1, and the last term can be found by multiplying the number of terms (n) by 2 and subtracting 1. In this case, n is 70, so the last term would be 70 * 2 - 1 = 139.
Now, you can plug these values into the formula:
Sum = 70/2 * (1 + 139)
= 35 * 140
= 4,900
Therefore, the sum of the first 70 consecutive odd numbers is 4,900.
In terms of the reasoning used to solve this problem, it involves deductive reasoning. Deductive reasoning is a logical process that uses general principles or rules to draw specific conclusions. In this case, the formula for the sum of an arithmetic series is a general rule used to find the sum of the given consecutive odd numbers.
b) For the second question, to find the sum of the first 1,400 consecutive odd numbers, you can follow a similar approach.
The first term is still 1, and the last term can be found by multiplying the number of terms (n) by 2 and subtracting 1. So, the last term would be 1,400 * 2 - 1 = 2,799.
Plugging these values into the formula:
Sum = 1,400/2 * (1 + 2,799)
= 700 * 2,800
= 1,960,000
Therefore, the sum of the first 1,400 consecutive odd numbers is 1,960,000.
In terms of reasoning, this also involves deductive reasoning as the formula for the sum of an arithmetic series is used to find the solution.