The first 3 term of an AP are x,2x+1 and 5x-1 find x and the sum of the first 10 terms
d = 2x+1-x=x+1
S10 = 10/2 (2*x+9(x+1))
= 5(11x+9)
= 55x+45
(2x+1)-x = (5x-1)-(2x+1)
Now find x
How to solve this math
To find the value of x, we can use the fact that the common difference between consecutive terms in an arithmetic progression (AP) remains constant.
Given that the first term is x, the second term is 2x+1, and the third term is 5x-1, we can set up the following equations:
2x+1 = x + d (equation 1)
5x-1 = 2x+1 + d (equation 2)
where d is the common difference.
Simplifying equation 1, we get:
2x+1 = x + d
x + 1 = d (equation 3)
Substituting equation 3 into equation 2, we have:
5x-1 = 2x+1 + x + 1
5x-1 = 3x+2
Subtracting 3x from both sides, we get:
2x - 1 = 2
Adding 1 to both sides, we get:
2x = 3
Dividing both sides by 2, we obtain:
x = 3/2 or 1.5
So, x is equal to 1.5.
Now, to find the sum of the first 10 terms of the AP, we can use the formula:
Sum = (n/2) * (first term + last term)
In this case, the first term is x and the 10th term is x + 9d (where d is the common difference).
Substituting the values we have:
First term = x = 1.5
Last term = x + 9d = 1.5 + 9(1) = 10.5 (since d = 1)
Using the sum formula, we get:
Sum = (10/2) * (1.5 + 10.5)
Sum = 5 * 12
Sum = 60
Therefore, the sum of the first 10 terms of the given AP is 60.
To find the value of x, we need to use the given information that the first 3 terms form an arithmetic progression (AP). In an AP, the common difference is the difference between any two consecutive terms.
Given:
First term (a) = x
Second term (a + d) = 2x + 1
Third term (a + 2d) = 5x - 1
Using the definition of an AP, we can set up the following equations:
a + d = 2x + 1 --------(1)
a + 2d = 5x - 1 --------(2)
Now, let's solve these equations to find the value of x:
From equation (1), we can express 'd' in terms of 'x':
d = (2x + 1) - (x) = x + 1
Substituting the value of 'd' into equation (2), we get:
a + 2(x + 1) = 5x - 1
a + 2x + 2 = 5x - 1
a - 3x = -3
Now, let's substitute the value of 'a' from equation (1) into the above equation:
x + 1 - 3x = -3
-2x + 1 = -3
-2x = -4
x = 2
Therefore, the value of x is 2.
To find the sum of the first 10 terms, we can use the formula for the sum of an arithmetic progression:
Sn = (n/2) * (first term + last term)
Given:
First term (a) = x = 2
Common difference (d) = 1
To find the last term, we need to find the tenth term (a + 9d):
T10 = a + 9d = 2 + 9(1) = 11
Now, let's substitute the values into the sum formula:
S10 = (10/2) * (2 + 11)
S10 = 5 * 13
S10 = 65
Therefore, the sum of the first 10 terms of the given arithmetic progression is 65.