The 5th and 10th terms of A. P are - 12 and 27 respectively. Find the A. P and its 15th term. (B) The first three terms of an A. P are x and the 10th term.
Using the basic definitions:
a + 4d = -12
a + 9d = 27
subtract them
5d = 39
d = 39/5 or 7.8
sub back in ....
a+4(7.8) = -12
a = -43.2
sequence is:
-43.2 , -35.4, -27.6, -19.8, -12, ...
term15 = a + 14 d
= .....
B) The first three terms of an A. P are x and the 10th term.
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(A) To find the first term (a) and the common difference (d) in an arithmetic progression (A.P.), we can use the formula:
a + 4d = -12 (equation 1)
a + 9d = 27 (equation 2)
To solve these equations, we can subtract equation 1 from equation 2:
(a + 9d) - (a + 4d) = 27 - (-12)
9d - 4d = 39
5d = 39
d = 39/5 = 7.8
Now, substitute the value of d into either equation to find the value of a:
a + 4(7.8) = -12
a + 31.2 = -12
a = -12 - 31.2
a = -43.2
Therefore, the A.P. is -43.2, -35.4, -27.6, ..., and the 15th term can be found using the formula:
T(n) = a + (n - 1)d
where T(n) is the nth term, a is the first term, d is the common difference, and n is the term number. Plugging in the values:
T(15) = -43.2 + (15 - 1)(7.8)
T(15) = -43.2 + 14(7.8)
T(15) = -43.2 + 109.2
T(15) = 66
Therefore, the 15th term of the A.P. is 66.
(B) In this case, we are given the first three terms as x, y, and z (where x is the first term, y is the second term, and z is the third term), and the 10th term is unknown.
For an A.P., the difference between consecutive terms is constant. So, we have:
y - x = z - y
Simplifying, we get:
2y = x + z
To find the value of the 10th term, let's represent it as "T(10)". We can use the formula:
T(n) = a + (n - 1)d
where T(n) is the nth term, a is the first term, d is the common difference, and n is the term number. Plugging in the values:
T(10) = x + (10 - 1)d
T(10) = x + 9d
Therefore, the 10th term of the A.P. can be represented as x + 9d.
To find the arithmetic progression (A.P.) and its 15th term, we'll start by using the given information that the 5th term is -12 and the 10th term is 27.
(A) Finding the common difference (d):
The general formula for the nth term of an arithmetic progression is:
an = a + (n - 1)d
Given that the 5th term (a5) is -12, we can substitute the values into the formula:
-12 = a + (5 - 1)d --> -12 = a + 4d ...(1)
Similarly, the 10th term (a10) is 27, so we have:
27 = a + (10 - 1)d --> 27 = a + 9d ...(2)
Now we have a system of equations (1) and (2) with two variables, a and d. We can solve this system of equations to find the values of a and d.
Subtracting equation (1) from equation (2), we get:
27 - (-12) = (a + 9d) - (a + 4d)
39 = 5d
Dividing both sides by 5, we find the common difference:
d = 39/5
d = 7.8
Substituting the value of d back into equation (1):
-12 = a + 4(7.8)
-12 = a + 31.2
a = -12 - 31.2
a = -43.2
So, the first term (a) of the arithmetic progression is -43.2, and the common difference (d) is 7.8.
Now, we can find the 15th term (a15):
a15 = a + (15 - 1)d
a15 = -43.2 + (14)(7.8)
a15 = -43.2 + 109.2
a15 = 66
Therefore, the 15th term (a15) of the arithmetic progression is 66.
(B) In this case, we are given that the first three terms of the arithmetic progression are x and the 10th term.
Let's assume that the first term (a) is x, the second term (a2) is y, and the 10th term (a10) is z.
Using the formula for the nth term of an A.P., we have:
a2 = a + (2 - 1)d --> y = x + d ...(3)
a10 = a + (10 - 1)d --> z = x + 9d ...(4)
Since we know that the first term (a) is x, we can substitute x back into equations (3) and (4):
y = x + d ...(3)
z = x + 9d ...(4)
Now we have a system of equations (3) and (4) with two variables, x and d. We can solve this system of equations to find the values of x and d.
Subtracting equation (3) from equation (4), we get:
z - y = (x + 9d) - (x + d)
z - y = x + 9d - x - d
z - y = 8d
Dividing both sides by 8, we find the common difference:
d = (z - y)/8
Now, we can substitute the value of d back into equation (3) to find x:
y = x + [(z - y)/8]
8y = 8x + (z - y)
9y = 8x + z
x = (9y - z)/8
Therefore, the value of the first term (x) is (9y - z)/8 and the common difference (d) is (z - y)/8.