If the 10th term of an Ap is -27 and the 5th term is -12
What will be the sum of its 25 terms ?
First, we need to find the common difference, denoted by 'd', of the arithmetic progression (AP).
We are given that the 10th term is -27 and the 5th term is -12.
To find the common difference, we can use the formula for the nth term of an AP:
an = a1 + (n-1)d
where an is the nth term, a1 is the first term, and d is the common difference.
Using the information given, we have:
-27 = a1 + (10-1)d ... (1)
-12 = a1 + (5-1)d ... (2)
Solve equations (1) and (2) simultaneously.
From equation (2), rearrange to express a1 in terms of d:
a1 = -12 - 4d ... (3)
Substitute equation (3) into equation (1):
-27 = (-12 - 4d) + (10-1)d
-27 = -12 - 4d + 10d - d
-27 = -12 + 6d
Rearrange the equation:
6d = -27 + 12
6d = -15
d = -15/6
Simplify d:
d = -5/2
Now, we know that d = -5/2.
To find the sum of 25 terms, we will use the formula for the sum of an AP:
Sn = (n/2)(2a1 + (n-1)d)
where Sn is the sum of the first n terms.
Substituting in the given values, we have:
S25 = (25/2)(2a1 + (25-1)d)
Simplify:
S25 = (25/2)(2(-12 - 4d) + (25-1)d)
Simplify further by substituting the value of d:
S25 = (25/2)(2(-12 - 4(-5/2)) + (25-1)(-5/2))
Simplify and calculate:
S25 = (25/2)(2(-12 + 10) - (24)(5/2))
Simplify further:
S25 = (25/2)(2(-2) - 60/2)
S25 = (25/2)(-4 - 30)
S25 = (25/2)(-34)
Multiply:
S25 = -425
Therefore, the sum of the 25 terms of the AP is -425.
To find the sum of the first 25 terms of an arithmetic progression (AP), we can use the formula for the nth term of an AP:
nth term = a + (n - 1)d
Where:
- nth term is the term number you want to find,
- a is the first term of the AP,
- n is the term number, and
- d is the common difference between the terms.
In this case, we are given that the 10th term is -27 and the 5th term is -12. Let's use these values to find the first term (a) and the common difference (d).
Using the given information:
10th term = a + (10 - 1)d = -27
5th term = a + (5 - 1)d = -12
Now we have two equations with two unknowns:
a + 9d = -27
a + 4d = -12
Subtracting the second equation from the first equation, we get:
a + 9d - (a + 4d) = -27 - (-12)
5d = -15
d = -3
Substituting the value of d into the second equation, we can find the value of a:
a + 4(-3) = -12
a - 12 = -12
a = 0
Now that we know the first term (a = 0) and the common difference (d = -3), we can find the sum of the first 25 terms of the AP using the sum formula:
Sum of the first n terms of an AP = n/2 * (2a + (n - 1)d)
Substituting the given values into the formula, we get:
Sum of the first 25 terms = 25/2 * (2(0) + (25 - 1)(-3))
= 25/2 * (-2 + 24(-3))
= 25/2 * (-2 - 72)
= 25/2 * (-74)
= -925
Therefore, the sum of the first 25 terms of the given arithmetic progression is -925.
To find the sum of the first 25 terms, we need to find the common difference (d) of the arithmetic progression (AP).
We are given the 10th term (a10) as -27 and the 5th term (a5) as -12.
The formula for finding the nth term (an) of an AP is:
an = a1 + (n - 1)d
Substituting the given values, we have:
-27 = a1 + (10 - 1)d ----(1)
-12 = a1 + (5 - 1)d ----(2)
Solving equations (1) and (2) simultaneously will give us the values of a1 and d.
From equation (1):
-27 = a1 + 9d
From equation (2):
-12 = a1 + 4d
Let's subtract equation (2) from equation (1):
-27 - (-12) = (a1 + 9d) - (a1 + 4d)
-27 + 12 = 9d - 4d
-15 = 5d
d = -3
Now that we have found the common difference, we can substitute it back into either equation (1) or (2) to find the value of a1.
Using equation (1):
-27 = a1 + 9(-3)
-27 = a1 - 27
a1 = 0
Therefore, the first term (a1) is 0 and the common difference (d) is -3.
To find the sum of the first 25 terms (S25), we can use the formula:
S25 = (n/2)(2a1 + (n - 1)d)
Substituting the values:
S25 = (25/2)(2(0) + (25 - 1)(-3))
S25 = 12(0 + 24(-3))
S25 = 12(0 - 72)
S25 = 12(-72)
S25 = -864
Therefore, the sum of the first 25 terms of the arithmetic progression is -864.