The sum of 16 terms of an AP is -504, while the sum of its 9 terms is -126. Find the sum of its 30 terms
Explain
The sum of 16 terms of an AP is -504
gives us: 8(2a + 15d) = -504
2a + 15d = - 63
the sum of its 9 terms is -126
gives us: (9/2)(2a + 8d) = -126
2a + 8d = - 28
subtract them,
7d = - 35
d = - 5 , then a = 6
sum(30) = 15(-10 + 29(6)) = ....
How did you get 6
Oh, my mathematical sideshow! It seems we have a puzzle on our hands. Let's navigate this AP circus together.
We know that the sum of 16 terms of the arithmetic progression (AP) is -504 and the sum of its 9 terms is -126.
To find the sum of 30 terms, we have to put those clown shoes on and solve this riddle step by step.
Let's start by finding the common difference, shall we?
The sum of 16 terms is -504, so we have 16/2 * (2a + (16-1)d) = -504.
Simplifying that equation, we get 8(2a + 15d) = -504.
Now, let's find the sum of 9 terms. We have 9/2 * (2a + (9-1)d) = -126.
Simplifying that equation, we get 4(2a + 8d) = -126.
Now that we have two equations, we can solve for 'a' and 'd.' Let's do it, Jester-style!
From the first equation, we get 2a + 15d = -63. And from the second equation, we get 2a + 8d = -31.
Now, let's subtract the second equation from the first equation: (2a + 15d) - (2a + 8d) = -63 - (-31).
Simplifying that expression, we get 7d = -32. Now, divide both sides by 7 to isolate 'd' and you'll find that d = -32/7.
Now that we have the common difference, we can go back to the first equation and find 'a.'
2a + 15(-32/7) = -63. Simplifying that equation, we get 2a - 480/7 = -63.
To make things simpler, let's move the constant over to the right side of the equation: 2a = -63 + 480/7.
Adding those two numbers, we get 2a = -441/7 + 480/7, which simplifies to 2a = 39/7.
Now, divide both sides by 2 to solve for 'a,' and you'll find that a = 39/14.
To find the sum of 30 terms, we can use the formula (n/2)(2a + (n-1)d).
Plugging in our values, we have (30/2)(2(39/14) + (30-1)(-32/7)).
Simplifying that expression, we get 15(39/7 - (29)(32/7)).
Let's do some more juggling: 15(39/7 - (928/7)).
Calculating that, we get 15(39/7 - 928/7), which simplifies to 15(-889/7).
The final answer, ladies and gentlemen of the mathematical circus, is -889/7.
So, the sum of 30 terms of this AP is -889/7. Ta-da!
To find the sum of 30 terms of an arithmetic progression (AP), we can use the formula for the sum of an AP:
Sn = (n/2)(2a + (n-1)d)
where Sn is the sum of the first n terms, 'a' is the first term of the AP, 'd' is the common difference, and 'n' is the number of terms.
Given that the sum of 16 terms is -504, we can substitute the values into the formula:
-504 = (16/2)(2a + (16-1)d)
-504 = 8(2a + 15d)
-63 = 2a + 15d -- equation 1
Similarly, given that the sum of 9 terms is -126, we can substitute the values into the formula:
-126 = (9/2)(2a + (9-1)d)
-126 = 4.5(2a + 8d)
-28 = 2a + 8d -- equation 2
We have two equations (equation 1 and equation 2) with two variables (a and d). We can solve these equations simultaneously to find the values of a and d.
By multiplying equation 2 by 4, we get:
-112 = 8a + 32d -- equation 3
By subtracting equation 3 from equation 1, we can eliminate variable 'a':
63 - (-112) = 2a + 15d - (8a + 32d)
175 = -6a - 17d
6a + 17d = -175 -- equation 4
Now, we can solve equations 4 and 2 simultaneously:
6a + 17d = -175 -- equation 4
2a + 8d = -28 -- equation 2
Multiplying equation 2 by 3, we get:
6a + 24d = -84 -- equation 5
By subtracting equation 5 from equation 4, we can eliminate variable 'a':
(6a + 17d) - (6a + 24d) = -175 - (-84)
-7d = -91
d = -91 / -7
d = 13
Substituting the value of 'd' into equation 2, we can find the value of 'a':
2a + 8(13) = -28
2a + 104 = -28
2a = -28 - 104
2a = -132
a = -66
Now that we have found the values of 'a' and 'd', we can substitute them into the formula for the sum of 30 terms:
S30 = (30/2)(2(-66) + (30-1)(13))
S30 = 15(-132 + 29(13))
S30 = 15(-132 + 377)
S30 = 15(245)
S30 = 3675
Therefore, the sum of the 30 terms in this arithmetic progression is 3675.