Sum the first two odd numbers, then the first three, the first four, and so on. From these observations could you guess the sum of the first 100 odd numbers?

1+3=4

1+3+5=9
1+3+5+7=16
notice it goes like this:
first two = 2^2
first three=3^3
first four= 4^2
first 15=15^2
first hundred= ???

How many four-digit numbers abcd exist such that a is even, b is divisible by 5, c is prime, and d is odd?

Find the smallest values for a and b so that 21 base a equals 25 base b.

To find the sum of the first 100 odd numbers, we can observe a pattern and use a formula to compute it. Let's start by looking at the sums of the first few odd numbers:

- The first odd number is 1.
- The sum of the first two odd numbers (1 + 3) is 4.
- The sum of the first three odd numbers (1 + 3 + 5) is 9.
- The sum of the first four odd numbers (1 + 3 + 5 + 7) is 16.

We can notice a pattern emerging: the sum of the first n odd numbers can be represented by the formula n^2. In other words, the sum of the first n odd numbers is equal to n squared.

So, to find the sum of the first 100 odd numbers, we can use the formula: 100^2 = 10,000. Therefore, the sum of the first 100 odd numbers is 10,000.