the sixth term of an arithmetic progression is 21 and the sum of the first 17 terms is 0. write the first three terms.
a+5d=21
17/2 (2a+16d) = 0
The sequence is
56, 49, 42, 35, 28, 21, 14, 7, 0, ...
Answer
Well, that's quite an interesting arithmetic progression, isn't it? Let's see if we can figure out the first three terms.
We know that the sixth term is 21, but unfortunately, that doesn't give us enough information to determine the first term, the common difference, or any of the other terms in between.
Now, the sum of the first 17 terms being 0 is quite perplexing. It seems like we have a bit of a mathematical contradiction going on here. Perhaps this is a perplexing puzzle designed to tease our brains.
In any case, since we are unable to determine the first three terms of the arithmetic progression given the information provided, we are left scratching our clown noses in confusion.
To find the first three terms of an arithmetic progression, we need to know the common difference (d). We can find the common difference by using the formula:
d = (aₙ - a₁) / (n - 1),
where aₙ is the nth term, a₁ is the first term, and n is the position of the desired term.
Since we are given the sixth term (a₆ = 21) and the sum of the first 17 terms is 0, we can use this information to find the common difference. The sum of an arithmetic progression is given by:
Sₙ = (n/2) * (2a₁ + (n-1)d),
where Sₙ is the sum of n terms.
In this case, we have:
S₁₇ = (17/2) * (2a₁ + (17-1)d) = 0,
which simplifies to:
17a₁ + 136d = 0.
Now we can solve the above equation simultaneously with the equation a₆ = 21 to find the first term (a₁) and the common difference (d). Substituting aₙ = a₁ + (n-1)d, we get:
a₁ + 5d = 21,
17a₁ + 136d = 0.
From the first equation, we can express the first term in terms of the common difference:
a₁ = 21 - 5d.
Substituting the value of a₁ in the second equation:
17(21 - 5d) + 136d = 0,
357 - 85d + 136d = 0,
357 + 51d = 0,
51d = -357,
d = -7.
Now we can find the first term (a₁):
a₁ = 21 - 5d,
a₁ = 21 - 5(-7),
a₁ = 21 + 35,
a₁ = 56.
Therefore, the first three terms of the arithmetic progression are: 56, 49, 42.
To solve this problem, we need to use the formulas for arithmetic progressions.
Let's call the first term of the arithmetic progression "a" and the common difference between the terms "d".
We are given that the 6th term of the arithmetic progression is 21. Using the formula for the nth term of an arithmetic progression, we can write:
a + 5d = 21 (since the sixth term is a + 5d)
We are also given that the sum of the first 17 terms is 0. Using the formula for the sum of an arithmetic progression, we can write:
17/2 * (2a + (17-1)d) = 0
Now we have two equations with two unknowns. We can solve these equations simultaneously to find the values of "a" and "d", which will give us the first three terms of the arithmetic progression.
Let's solve the equations:
Equation 1: a + 5d = 21
Equation 2: 17/2 * (2a + 16d) = 0
From Equation 1, we can rewrite it as:
a = 21 - 5d
Substituting this into Equation 2, we get:
17/2 * (2(21 - 5d) + 16d) = 0
17/2 * (42 - 10d + 16d) = 0
17/2 * (42 + 6d) = 0
17 * (42 + 6d) = 0
714 + 102d = 0
102d = -714
d = -714/102
d = -7
Now that we have found the value of "d", we can substitute it back into Equation 1 to find the value of "a":
a + 5(-7) = 21
a - 35 = 21
a = 21 + 35
a = 56
Therefore, the first term (a) is 56 and the common difference (d) is -7.
The first three terms of the arithmetic progression are:
a = 56
a + d = 56 + (-7) = 49
a + 2d = 56 + 2(-7) = 42
So, the first three terms of the arithmetic progression are 56, 49, and 42.