A sequence of numbers begins with 1, -1, -1, 1, -1. Each new number is found by taking the product of the
two preceding numbers. For instance the sixth number is the product of the fourth and fifth numbers. What is
the sum of the first 2013 numbers?
well, just look at the sequence: 1,-1,-1, 1,-1,-1, 1,-1,-1,...
You can see that it consists of groups of 3: 1,-1,-1
Each group has sum -1
2013 = 3*671
So the sum will be -671
To find the sum of the first 2013 numbers in the given sequence, we will need to calculate each number in the sequence and then add them together.
We are given that the first five numbers in the sequence are: 1, -1, -1, 1, -1.
To find the next number, we take the product of the two preceding numbers. Therefore, the sixth number is (-1) * (-1) = 1.
Continuing this pattern, we can find the next numbers in the sequence:
Seventh number = (-1) * (1) = -1
Eighth number = (1) * (-1) = -1
Ninth number = (-1) * (-1) = 1
The pattern repeats as 1, -1, -1, 1, -1, 1, -1, ...
Since the sequence alternates between 1 and -1, we can determine the value of the nth number in the sequence by looking at the parity (whether the integer is even or odd) of n.
If n is even, the number is 1. If n is odd, the number is -1.
To find the sum of the first 2013 numbers in the sequence, we need to determine how many numbers are 1 and how many are -1.
Since the sequence repeats every two numbers, every even positive integer is 1 and every odd positive integer is -1.
Since 2013 is an odd number, there are 1006 even numbers and 1007 odd numbers.
Thus, the sum of the first 2013 numbers in the sequence is:
1006 * 1 + 1007 * (-1) = 1006 - 1007 = -1.
So, the sum of the first 2013 numbers in the given sequence is -1.