Question

what conjecture can you make about the sum of the first 10 odd integers

Answer 1

Look at the sum of the first two odd integers, 1+3 = 4
Now, the first three odd integers, 1+3+5 = 9
Now, the first four odd integers, 1+3+5+7 = 16
What do you notice about these sums, 4,9,16? Anything special about these numbers?

Now, what do you think would be a good conjecture for the sum of the first 10 odd integers?

Answer 2

The sum \(S_n\) of first n positive odd integers is given by \(n^2\). This is one of the simplest relationship between the sum of n terms of a sequence and the number of terms.

This \(S_n\) for first n positive odd integers can be verified by considering the fact that the sum of numbers of an arithmetic sequence is given by \(\dfrac{n}{2[(a+L)d]}\)

where n = number of terms.

a = first term

L = last term and

d = common difference

For consecutive positive odd integers, the common difference is equal to 2.

Answer 3

Conjecture is the formation or expression of an opinion or theory without sufficient evidence for proof. Conjecture on the sum of the first 10 positive, even numbers is that its sum would be larger than zero. \