The first 3 terms of an arithmetic progression are x,2x+1,5x-1 find x and the sum of the first 10th terms
To find the common difference in an arithmetic progression, we subtract the second term from the first term:
Common Difference = (2x + 1) - x
Common Difference = x + 1
Now, let's use the common difference to find the third term. We add the common difference to the second term:
Third Term = (5x - 1) + (x + 1)
Third Term = 6x
Since the third term is given as 6x, we can equate it to the expression we got earlier:
6x = 5x - 1
To solve for x, we subtract 5x from both sides of the equation:
6x - 5x = -1
x = -1
So, x is equal to -1. Now we can substitute this value back to find the first term and the common difference:
First Term = x = -1
Common Difference = x + 1 = -1 + 1 = 0
The first term is -1, and the common difference is 0. This means our arithmetic progression is -1, -1, -1, -1, ... with a common difference of 0.
To find the sum of the first 10 terms, we can use the formula for the sum of an arithmetic progression:
Sum = (n/2) * (2a + (n - 1)d)
where n is the number of terms, a is the first term, and d is the common difference.
Using our values:
n = 10
a = -1
d = 0
Sum = (10/2) * (2(-1) + (10 - 1)0)
Sum = 5 * (-2)
Sum = -10
Therefore, the sum of the first 10 terms of the arithmetic progression is -10.