if 3,u,v,24 are in AP, find u+v
pls help me solve it
think about it. Consider any 4 consecutive terms of any AP:
a, a+d, a+2d, a+3d
The sum of the two middle terms (2a+3d) is the same as the sum of the two end terms (2a+3d).
You told me a = 3 and
a+ 3d = 24
3d = 21
d = 7
so your sequence is 3, 10, 17, 24
clearly u = 10 and v = 17
so u+v = 27
To find the value of u+v, we need to determine the common difference of the arithmetic progression (AP) and then apply the property of an AP that states every successive term is obtained by adding the common difference to the preceding term.
Given the elements of the AP: 3, u, v, 24
To find the common difference, we can take the difference between any two consecutive terms in the AP. Let's consider the difference between the second and first terms:
u - 3
Similarly, let's consider the difference between the third and second terms:
v - u
As it is an AP, these two differences should be equal. So, we can equate them:
u - 3 = v - u
Simplifying the equation, we get:
2u = v + 3
Now, let's consider the difference between the fourth and third terms:
24 - v
Again, according to the AP property, this should be equal to the difference between the third and second terms:
v - u
Equating these two differences:
v - u = 24 - v
Simplifying the equation, we get:
2v = u + 24
Now, we can solve the system of equations formed by these two equations:
2u = v + 3
2v = u + 24
Rearranging the first equation, we get:
v = 2u - 3
Substituting this value of v in the second equation, we have:
2(2u - 3) = u + 24
Expanding and simplifying the equation, we get:
4u - 6 = u + 24
Bringing all the terms involving u to one side, we have:
4u - u = 24 + 6
Simplifying further, we get:
3u = 30
Dividing both sides by 3, we have:
u = 10
Now, substituting this value of u back into the equation v = 2u - 3, we get:
v = 2*10 - 3
v = 20 - 3
v = 17
Therefore, the values of u and v are:
u = 10
v = 17
To find the value of u+v, we can simply add these two:
u + v = 10 + 17
u + v = 27
Hence, u + v = 27.