The first three terms of an AP are x ,2x+1 and 5x-1.Find the value of x and the sum of the first 10 terms
I don't really understand
Solve in written form for clear understanding
To find the value of x, we need to identify the common difference between the terms of the arithmetic progression (AP). We can do this by subtracting the second term from the first term and the third term from the second term.
Let's subtract the second term (2x+1) from the first term (x):
x - (2x+1) = x - 2x - 1 = -x - 1
Now, let's subtract the third term (5x-1) from the second term (2x+1):
(2x+1) - (5x-1) = 2x + 1 - 5x + 1 = -3x + 2
Since both of these differences represent the same common difference, we can equate them:
-3x + 2 = -x - 1
Let's solve this equation to find the value of x:
-3x + x = -1 - 2
-2x = -3
x = -3 / -2
x = 3/2
x = 1.5
Therefore, the value of x is 1.5.
To find the sum of the first 10 terms of the AP, we can use the formula for the sum of an AP:
Sum of AP = (n / 2) * (2a + (n - 1) * d),
where n is the number of terms, a is the first term, and d is the common difference.
In this case, we know that a = x (the first term), d = -x - 1 (the common difference), and n = 10 (the number of terms).
Let's substitute these values into the formula to find the sum of the first 10 terms:
Sum = (10 / 2) * (2x + (10 - 1) * (-x - 1))
= 5 * (2x + 9(-x - 1))
= 5 * (2x - 9x - 9)
= 5 * (-7x - 9)
= -35x - 45
Now, we'll substitute the value of x:
Sum = -35(1.5) - 45
= -52.5 - 45
= -97.5
Therefore, the sum of the first 10 terms of the AP is -97.5.
since there is a common difference,
2x+1 - x = 5x-1 - (2x+1)
Then just use your formula for S10