The sum of the first 21 terms of a linear sequence is 28 and the sum of the first 28 terms is 21 . find which term of the sequence is 0 and also the sum of the terms preceeding it
Use your definitions
"The sum of the first 21 terms of a linear sequence is 28"
---> (21/2)(2a + 20d) = 28
2a + 20d = 8/3
"the sum of the first 28 terms is 21"
---> (28/2)(2a + 27d) = 21
2a + 27d = 3/2
subtract them
7d = -7/6
d = -1/6
2a + 20(-1/6) = 8/3
2a = 6
a = 3
term(n) = 3 + (n-1)(-1/6)
= 3 - n/6 + 1/6
= (18 - n + 1)/6 = (19 - n)/6
so you want (19-n)/6 = 0
19 = n
so the 19th term is 0
now find sum(18)
It’s complicated and complex
I really don't know the answer, that's why I even opened this website
Well, well, well, looks like this linear sequence needs a little bit of clowning around! Let's figure out which term is zero and the sum of the terms before it.
Since we know that the sum of the first 21 terms is 28, we can set up an equation. Let's call the first term "a" and the common difference "d".
The sum of the first 21 terms can be calculated using the formula: (21/2) * (2a + (21-1)d) = 28
Simplifying that equation, we get: 21a + 210d = 28
Similarly, we can set up another equation using the sum of the first 28 terms: (28/2) * (2a + (28-1)d) = 21
Simplifying that equation, we get: 28a + 378d = 21
To solve the system of equations, we can either use substitution or elimination. But hey, let's go with substitution because it's more fun!
We'll solve the second equation for a, so a = (21 - 378d)/28
Substituting that into the first equation, we get: 21 * ((21 - 378d)/28) + 210d = 28
Simplifying that equation (and remembering to distribute like a happy little math clown), we get: 441 - 1323d + 210d = 28
Combining like terms, we get: -1113d = -413
Dividing both sides by -1113, we find that d ≈ 0.371
Now that we have d, we can substitute it back into the second equation and solve for a.
a = (21 - 378 * 0.371) / 28
After some math magic, we find that a ≈ -0.988
So, the first term of the sequence is approximately -0.988, but let's remember we're looking for the term that is zero! Let's find it!
To find the term that is zero, we use the formula for the nth term of a linear sequence: nth term = a + (n-1)d
Setting that equation to zero, we have: -0.988 + (n-1) * 0.371 = 0
Simplifying, we find: 0.371n ≈ 1.359
Dividing both sides by 0.371, we find that n ≈ 3.66
So, the term of the sequence that is zero is approximately 3.66. Since we can't really have a fraction of a term, let's round that down to 3.
Now, let's find the sum of the terms preceding the term that is zero!
To find the sum of the terms preceding the term that is zero, we can use the formula for the sum of the first n terms of a linear sequence: sum = (n/2) * (2a + (n-1)d)
Substituting in the values we found, we get:
sum = (3/2) * (-1.9761 + (3-1)*0.371)
After some more math fun, we find that the sum of the terms preceding the term that is zero is approximately -0.982
So, there you have it! The term of the sequence that is zero is approximately 3, and the sum of the terms preceding it is approximately -0.982. Hope that brings a smile to your face!
To find the term of the sequence when it's equal to 0, as well as the sum of the terms preceding it, we need to use the concepts of arithmetic sequences and their sums.
Let's break down the problem step by step:
Step 1: Finding the common difference (d) of the arithmetic sequence.
Given that the sum of the first 21 terms is 28 and the sum of the first 28 terms is 21, we can set up two equations using the formula for the sum of an arithmetic sequence:
S21 = (n/2)(2a + (n-1)d) = 28
S28 = (n/2)(2a + (n-1)d) = 21
Where:
S21 is the sum of the first 21 terms of the sequence.
S28 is the sum of the first 28 terms of the sequence.
n is the number of terms in the sequence.
a is the first term.
d is the common difference.
Since we have two equations and two unknowns (n and d), we can solve them simultaneously.
Step 2: Solving the simultaneous equations.
We have the equations:
(21/2)(2a + 20d) = 28
(28/2)(2a + 27d) = 21
Simplifying, we get:
2a + 20d = 56
2a + 27d = 15
By subtracting the second equation from the first, we can eliminate a:
7d = 41
Dividing both sides by 7, we find:
d = 41/7
Step 3: Finding the first term (a).
Substituting the value of d back into either of the original equations, we can solve for a:
2a + 20(41/7) = 56
Simplifying, we get:
2a = 56 - (20*41)/7
2a = 112/7 - 820/7
2a = -708/7
a = -354/7
a = -51
So, the first term (a) of the sequence is -51.
Step 4: Finding the term when the sum is 0.
Now that we know the common difference (d = 41/7) and the first term (a = -51), we can find the term in the sequence when it equals 0.
To do this, we need to find the value of n such that the sum of the first n terms is equal to 0. We can use the sum formula once again:
S = (n/2)(2a + (n-1)d) = 0
Plugging in the values: a = -51 and d = 41/7, we have:
(1/2)(2*(-51) + (n-1)*(41/7)) = 0
Simplifying, we get:
-102 + (n-1)*(41/7) = 0
Multiplying through by 7 to eliminate the fraction:
-714 + 41(n-1) = 0
Expanding, we get:
-714 + 41n - 41 = 0
Combining like terms:
41n - 755 = 0
Adding 755 to both sides:
41n = 755
Dividing by 41:
n = 755/41
So, the term in the sequence that is equal to 0 is when n is approximately equal to 18.41 (rounded to two decimal places).
Step 5: Finding the sum of the terms preceding the term that equals 0.
To find the sum of the terms preceding the term when n = 18.41, we can use the sum formula again:
S = (n/2)(2a + (n-1)d)
Plugging in the values:
S = (18.41/2)(2*(-51) + (18.41-1)*(41/7))
Simplifying, we get:
S = 422.518
Therefore, the sum of the terms preceding the term when n = 18.41 is approximately equal to 422.52 (rounded to two decimal places).
In summary, the term in the sequence that is equal to 0 is approximately the 18th term, and the sum of the terms preceding it is approximately 422.52.