In an ap the5 th and 10 th terms are in ratio 1:2 and the 12th term is 36 find a and d
(a+4d)/(a+9d) = 1/2
a+11d = 36
Now you can find a and d.
please tell me
To find the values of 'a' and 'd' in an arithmetic progression (AP), we can use the given information.
Let's start by finding the common difference 'd' of the AP.
Given that the ratio of the 5th term to the 10th term is 1:2, we can write it as:
a + 4d : a + 9d = 1 : 2
To simplify, we can multiply both sides of the equation by 2:
2(a + 4d) : 2(a + 9d) = 1 : 2
2a + 8d : 2a + 18d = 1 : 2
Now, we can equate the corresponding terms:
2a + 8d = 1
2a + 18d = 2
Subtracting the first equation from the second equation eliminates the 'a' term:
2a + 18d - (2a + 8d) = 2 - 1
10d = 1
Now we have the value of 'd'. We can solve for 'a' using the given information:
The 12th term of the AP is 36.
We know that the 12th term can be expressed as:
a + 11d = 36
Substituting the value of 'd' we found earlier:
a + 11(1/10) = 36
a + 11/10 = 36
Multiply both sides by 10 to eliminate fractions:
10a + 11 = 360
Subtract 11 from both sides:
10a = 360 - 11
10a = 349
Divide both sides by 10:
a = 34.9
Therefore, in the given AP, 'a' is approximately 34.9 and 'd' is 1/10.
To find the values of 'a' and 'd' in an arithmetic progression (AP), we can use the information given.
Let's start by considering the given ratio of the 5th and 10th terms:
1st ratio = 1:2
In an AP, we know that the nth term can be expressed as:
an = a + (n - 1)d
For the 5th term, we have:
a5 = a + 4d
For the 10th term, we have:
a10 = a + 9d
Given the ratio of the 5th and 10th terms as 1:2, we can set up the following equation:
(a + 4d) / (a + 9d) = 1 / 2
Now, let's work with the equation for the 12th term:
a12 = a + 11d
Given that the 12th term is 36, we have:
a + 11d = 36
Now we have two equations:
(a + 4d) / (a + 9d) = 1 / 2 -- (equation 1)
a + 11d = 36 -- (equation 2)
To solve these equations, we can use substitution. Let's rearrange equation 2 to express 'a' in terms of 'd':
a = 36 - 11d
Substituting this value for 'a' in equation 1, we have:
((36 - 11d) + 4d) / ((36 - 11d) + 9d) = 1 / 2
Now, solve this equation for 'd'.
((36 - 11d) + 4d) / (36 - 2d) = 1 / 2
Multiplying both sides of the equation by 2(36 - 2d):
(36 - 11d + 4d) * 2 = 36 - 2d
72 - 22d = 36 - 2d
-22d + 2d = 36 - 72
-20d = -36
Dividing both sides of the equation by -20:
d = -36 / -20
d = 1.8
Now that we have the value of 'd', we can substitute it back into equation 2 to find 'a':
a + 11(1.8) = 36
a + 19.8 = 36
a = 36 - 19.8
a = 16.2
Therefore, the values of 'a' and 'd' in the arithmetic progression are:
a = 16.2 and d = 1.8.