find the values of p and q in the arithmetic progression: -12,p,q,18.
q - p = d
18 - q = d
p + 12 = d
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so q = 18-d and p = d - 12
so (18-d) - ( d-12) = d
30 - 2 d = d
d = 10
q = 18 - 10 = 8
p = 8 -10 = -2
so
-12 , -2 , 8 , 18
the common difference is 10
... there are three differences from -12 to 18
... (18 - -12) / 3 = 10
-12 , -2 , 8 , 18
Correct
Correct
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Well, in an arithmetic progression, the difference between any two consecutive terms is always the same. So, we can find the common difference by subtracting any two consecutive terms.
Let's subtract the second term (-12) from the first term (p):
p - (-12) = p + 12
Now, let's subtract the third term (q) from the second term (-12):
-12 - q = -12 - q
Finally, let's subtract the fourth term (18) from the third term (q):
q - 18 = q - 18
Since we know that the common difference is the same for all these equations, we can set them equal to each other:
p + 12 = -12 - q = q - 18
Simplifying, we get:
p + 12 = -12 - q,
p + 12 + q = -12,
p + q = -24.
So, the values of p and q can be any two numbers that add up to -24. There are multiple possible options, such as p = -6 and q = -18, or p = -12 and q = -12, or p = -24 and q = 0, just to name a few.
Remember, finding the values of p and q is not unique and there could be many possible solutions!
To find the values of p and q in the arithmetic progression -12, p, q, 18, we can use the formula for the nth term of an arithmetic progression. The formula is:
an = a1 + (n-1)d
where an represents the nth term, a1 is the first term, d is the common difference, and n is the position of the term.
Here, we know that the first term a1 is -12, and the fourth term, a4, is 18. In an arithmetic progression, the common difference (d) is the same between any two consecutive terms. So, we can write:
a4 = a1 + (4-1)d
Substituting the values we have, we get:
18 = -12 + 3d
Now, let's solve this equation to find the value of d (the common difference):
18 = -12 + 3d
30 = 3d
d = 10
Now that we know the common difference is 10, we can find the values of p and q.
The second term, a2, is given by:
a2 = a1 + (2-1)d
a2 = -12 + 1*10
a2 = -12 + 10
a2 = -2
So, the value of p is -2.
Similarly, the third term, a3, is given by:
a3 = a1 + (3-1)d
a3 = -12 + 2*10
a3 = -12 + 20
a3 = 8
Therefore, the value of q is 8.
In conclusion, the values of p and q in the given arithmetic progression -12, p, q, 18 are p = -2 and q = 8.