Given an arithmetic progression -7,-3,1,..., state three consecutive terms in this progression which sum up to 75.
+ 4 = change from term to term
x-4 + x + x+4 = 75
3 x = 75
x = 25
so
21, 25, 29
Think about how you can locate these three consecutive terms.
What is the first term, and the common difference?
The sum of an AP for the first n terms is given by
S(n) = (n/2)(2a+(n-1)d)
where a = -7, d = 4
So lets say the 3 consecutive terms are the 14,15 and 16th term.
to get the sum of 75 of these 2 terms, i take
S(16) - S(13)
hence,
S(n) - (n-3) = 75
substituting the formula,
(n/2)(2a+(n-1)d) - [(n-3)/2][2a+(n-3-1)d] = 75
solve and find n.
To find three consecutive terms in the given arithmetic progression that sum up to 75, we can solve this problem by setting up an equation.
Let's assume that the common difference is represented by 'd', and the first term of the progression is 'a'.
According to the given arithmetic progression, we can determine the values of 'a' and 'd'.
The first term, 'a', is -7, and the second term is -3.
a = -7
Second term = a + d = -3
By substituting the value of a into the second equation, we get:
-7 + d = -3
d = -3 + 7
d = 4
Now, we can write the explicit formula for the nth term of the arithmetic progression:
Tn = a + (n - 1) * d
To find the three consecutive terms that sum up to 75, we can set up the equation:
Tn + T(n + 1) + T(n + 2) = 75
Substituting the values obtained above:
(-7 + (n - 1) * 4) + (-7 + (n) * 4) + (-7 + (n + 1) * 4) = 75
Simplifying the equation, we get:
-21 + 12n = 75
Rearranging the equation, we have:
12n = 75 + 21
12n = 96
n = 8
Now, let's find the three consecutive terms:
T8 = -7 + (8 - 1) * 4 = -7 + 7 * 4 = 21
T9 = -7 + (9 - 1) * 4 = -7 + 8 * 4 = 25
T10 = -7 + (10 - 1) * 4 = -7 + 9 * 4 = 29
Therefore, the three consecutive terms in this arithmetic progression that sum up to 75 are 21, 25, and 29.
To find three consecutive terms in this arithmetic progression that sum up to 75, we need to first determine the common difference of the progression.
The arithmetic progression -7, -3, 1,... represents the formula:
an = a1 + (n - 1)d
where "an" represents the value of the nth term, "a1" is the first term, and "d" is the common difference.
From the given progression: -7, -3, 1,...
We can see that the common difference is 4, as there is an increase of 4 between each consecutive term.
Now, let's assume that the three consecutive terms we are looking for are:
n, n + 4, n + 8
According to the question, the sum of these three terms is 75:
n + (n + 4) + (n + 8) = 75
Simplifying the equation, we get:
3n + 12 = 75
Subtracting 12 from both sides:
3n = 63
Dividing both sides by 3:
n = 21
Therefore, the three consecutive terms in the progression -7, -3, 1,... that sum up to 75 are:
21, 25, 29.