The sum of the first three consecutive terms in AP is 21 and the last to has ratio 6:1. Find the numbers
The phrase " .... and the last to has ratio 6:1 " makes no sense.
Anonymous assumed the ratio of 6:1 was the third term to 2nd term, I don't know why.
If you meant that the ratio of the last two terms was 6:1, then
we could get
(a+d)/(a+2d) + 6 , depending on who's first..
a + a+d + a+2d = 21
3 a + 3 d = 21
a+d = 7
d = 7-a
(a+2d) / (a+d) =6
(a+ 14-2a) / 7 = 6
(14-a) = 42
a = - 28
d = 7+28 = 35
-28 , 7 , 42
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check
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sum = 21
42/7 = 6 sure enough
I assumed that the w was missing from
" .... last tWo has ratio ..."
To find the numbers in the arithmetic progression (AP), we need to use the given information and solve for the unknown terms.
Let's assume that the first term of the AP is "a," and the common difference is "d."
Given:
The sum of the first three consecutive terms in the AP is 21.
So, using the formula for the sum of an arithmetic series, we have:
Sum = (Number of terms / 2) * (2 * First term + (Number of terms - 1) * Common difference)
Substituting the values:
21 = (3/2) * (2a + 2d)
Simplifying the equation:
21 = (3/2) * (2a + 2d)
42 = 3(2a + 2d)
42 = 6a + 6d
6a + 6d = 42 ---(Equation 1)
The ratio of the last two terms is 6:1, which means the second-to-last term (a + d) is 6 times the last term (a + 2d).
So, we have:
(a + d) / (a + 2d) = 6 / 1
Cross multiplying, we get:
(a + d) = 6(a + 2d)
a + d = 6a + 12d
d - 11d = 6a - a
-10d = 5a
5a = -10d ---(Equation 2)
We now have a system of two equations (Equation 1 and Equation 2) with two unknowns (a and d), which we can solve simultaneously.
Substituting Equation 2 into Equation 1, we get:
6a + 6d = 42
6(-10d) + 6d = 42
-60d + 6d = 42
-54d = 42
d = 42 / -54
d = -7/9
Now, substitute the value of d back into Equation 2 to find a:
5a = -10(-7/9)
5a = 70/9
a = (70/9) / 5
a = 14/9
Therefore, the first three consecutive terms in the arithmetic progression are:
a = 14/9
a + d = 14/9 + (-7/9) = 7/9
a + 2d = 14/9 + (-14/9) = 0/9 = 0
So, the numbers in the arithmetic progression are 14/9, 7/9, and 0.