the first three terms of an a.p are x,2x+1 and 5x-1.find x and the sum of the first 10 terms
2x-1;3x;5x-2
5x-2-3x=3x-(2x-1)
x=3
S = ½(2a + (n-1)d)n
5*3-2=13
3x=9
13-9=4
2x+1 - x = 5x-1 - (2x+1)
find x. Now you know that a=x and d=x+1
S10 = 10/2 (2a+9d)
To find the value of x, we can use the fact that the terms of an arithmetic progression (AP) have a common difference.
Given the first term x, the second term is 2x + 1, and the third term is 5x - 1.
To find the common difference, we can subtract the second term from the first term or the third term from the second term.
Second term - first term: (2x + 1) - x = x + 1
Third term - second term: (5x - 1) - (2x + 1) = 3x - 2
Since these common differences should be equal, we can set them equal to each other and solve:
x + 1 = 3x - 2
Now, let's solve this equation for x:
x - 3x = -2 - 1
-2x = -3
x = (-3) / (-2)
x = 3/2
So, the value of x is 3/2.
Now, to find the sum of the first 10 terms of the AP, we can use the formula for the sum of an AP:
S = (n/2) * (2a + (n - 1)d),
where:
S is the sum of the AP,
n is the number of terms in the AP,
a is the first term of the AP, and
d is the common difference of the AP.
In this case, we are given the first term a = x = 3/2, the number of terms n = 10, and the common difference d = (2x + 1) - x = 1.
Substituting the values into the formula, we get:
S = (10/2) * (2(3/2) + (10 - 1)(1))
= (5) * (3 + 9)
= (5) * (12)
= 60
Therefore, the sum of the first 10 terms of the arithmetic progression is 60.
To find the value of 'x' and the sum of the first 10 terms of an arithmetic progression (A.P.) given the first three terms, follow these steps:
Step 1: Identifying the terms of the A.P.
Given terms of the A.P. are:
First term (a1) = x
Second term (a2) = 2x + 1
Third term (a3) = 5x - 1
Step 2: Finding the common difference (d)
In an A.P., the common difference (d) is the constant difference between consecutive terms. To find 'd,' we subtract any two consecutive terms. Here, we'll subtract a2 - a1:
(2x + 1) - x = 2x - x + 1 = x + 1
So, the common difference (d) is x + 1.
Step 3: Setting up equations
Using the formula for the nth term of an A.P., we can set up two equations to find 'x.'
- Equations based on the first and second terms (a1 and a2):
a1 = x
a2 = a1 + d --> 2x + 1 = x + (x + 1)
- Equations based on the second and third terms (a2 and a3):
a2 = 2x + 1
a3 = a2 + d --> 5x - 1 = 2x + 1 + (x + 1)
Step 4: Solving the equations and finding 'x'
Let's solve the equations one by one:
From the first equation, we have: 2x + 1 = x + (x + 1)
Simplifying: 2x + 1 = 2x + 1
This equation is true for any value of x. So, 'x' can be any real number.
Step 5: Finding the sum of the first 10 terms
The formula to find the sum of the first 'n' terms of an A.P. is:
Sn = (n/2) * [2a1 + (n - 1) * d]
Given: n = 10, a1 = x, and d = x + 1
Substituting the values in the formula:
S10 = (10/2) * [2x + (10 - 1)(x + 1)]
Simplifying:
S10 = 5 * (2x + 9x)
S10 = 5 * 11x
S10 = 55x
Therefore, the sum of the first 10 terms is 55x.