The. Second,third,and fourth terms of an ap are ( x-2),5,(X+2) find the value of X and nineth term
To find the value of x, we can use the fact that the common difference between the terms of an arithmetic progression (AP) is constant.
From the given information, we have the second term = 5 and the first term = (x-2). Therefore, the common difference (d) can be found by subtracting the first term from the second term:
5 - (x-2) = 7 - x
Similarly, we can find the common difference between the third and second terms:
(x+2) - 5 = (x-3)
Since the common difference is the same, we can set these two expressions equal to each other:
7 - x = x - 3
Simplifying the equation, we have:
2x = 10
Dividing both sides by 2, we get:
x = 5
Therefore, the value of x is 5.
To find the ninth term, we can use the formula for the nth term of an arithmetic progression:
Tn = a + (n-1)d
where Tn represents the nth term, a is the first term, n is the position of the term, and d is the common difference.
In this case, the first term (a) is (x-2) = 5-2 = 3, and the common difference (d) is 7 - 5 = 2.
Substituting these values into the formula, we get:
T9 = 3 + (9-1)*2
= 3 + 8*2
= 3 + 16
= 19
Therefore, the ninth term of the arithmetic progression is 19.
sorry -- to find the common difference d, you need
5 - (x-2) = (x+2)-5
x=5, so d = 2
and thus, a = 1
and the sequence starts 1,3,5,7,...
a9 = 1+8*2 = 17
Apologies for the mistake in my previous response.
Correctly solving for the common difference, we have:
5 - (x-2) = (x+2)-5
Expanding the equation:
5 - x + 2 = x + 2 - 5
Combining like terms:
7 - x = x - 3
Rearranging the equation:
2x = 10
Dividing both sides by 2:
x = 5
Therefore, the value of x is indeed 5.
Since the common difference (d) is 2, the first term (a) can be found by substituting x=5 into the equation for the first term:
a = (x-2) = 5-2 = 3
Thus, the arithmetic progression starts with the sequence 1, 3, 5, 7, ...
To find the ninth term (a9), we can use the formula for the nth term of an arithmetic progression:
Tn = a + (n-1)d
Substituting the values into the formula:
a9 = 3 + (9-1)*2
= 3 + 8*2
= 3 + 16
= 19
Therefore, the ninth term of the arithmetic progression is 19.