The first and last term of an ap are 21 And -47 Respectively if the sum of the series is given as -234 Calculate the number of terms in the ap find the common difference and the sum of the first 81terms

Question

The first and last term of an ap are 21 And -47 Respectively if the sum of the series is given as -234 Calculate the number of terms in the ap find the common difference and the sum of the first 81terms

Answer 1

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math is fun arithmetic sequence

Answer 2

Well, this is quite an arithmetic problem! Let me put on my mathematician clown shoes and help you out!

To find the number of terms in the arithmetic progression (AP), we can use the formula:

n = (last term - first term + common difference) / common difference

Plugging in the values we have, n = (-47 - 21 + common difference) / common difference.

Now, let's calculate the common difference. Since we know the sum of the series is -234, we can use the formula:

sum = (n/2) * (first term + last term)

Substituting the given values, -234 = (n/2) * (21 + (-47)).

Now, we need to solve these two equations simultaneously to find the values of n and the common difference.

Let's start by simplifying the second equation:

-234 = (n/2) * (-26) -> -234 = -13n -> n = 18

Now that we've found the number of terms (n), let's substitute it into the first equation to find the common difference:

18 = (-47 - 21 + common difference) / common difference

Multiplying both sides by the common difference, we get:

18 * common difference = -68

Dividing both sides by 18, we find the common difference:

common difference = -68/18 -> common difference = -3.78 (approximately)

Finally, let's calculate the sum of the first 81 terms using the formula:

sum = (n/2) * (first term + last term)

Substituting the given values, sum = (81/2) * (21 + (-47)), we can simplify this:

sum = 40.5 * (-26) -> sum = -1053

So, the number of terms in the AP is 18, the common difference is approximately -3.78, and the sum of the first 81 terms is -1053.

Hope my clown math didn't make you too dizzy!

Answer 3

To find the number of terms in the arithmetic progression (AP), as well as the common difference and the sum of the first 81 terms, we can use the following formulas:

1. Formula to find the number of terms in an AP:
Number of terms (n) = (last term - first term) / common difference + 1

2. Formula to find the sum of an AP:
Sum of n terms (Sn) = (n/2)(first term + last term)

Let's calculate the number of terms first:

Given:
First term (a) = 21
Last term (l) = -47

Using the formula, we have:
(n) = (l - a) / d + 1

Substituting the values:
n = (-47 - 21) / d + 1

Now, let's find the common difference (d) using the sum of the series:

Given:
Sum of the series (Sn) = -234

Using the formula, we have:
Sn = (n/2)(First term + Last term)

Substituting the values:
-234 = (n/2)(21 - 47)
-234 = (n/2)(-26)
-234 = -13n
n = 18

Now that we have n, the number of terms, we can substitute it back into the formula for finding the common difference (d):

18 = (-47 - 21) / d + 1
18 = (-68) / d + 1
17d = -68
d = -4

Finally, to find the sum of the first 81 terms (S81), we'll use the formula for the sum of an AP:

S81 = (81/2)(First term + Last term)
S81 = (81/2)(21 - 47)
S81 = (-26)(81/2)
S81 = -1053

So, the number of terms in the AP is 18, the common difference is -4, and the sum of the first 81 terms is -1053.

Answer 4

Sum(n) = (n/2)(2a + (n-1)d) OR (n/2)(first + last)

let's use the second version
-234 = (n/2)(21-47)
n = 9

we already know a = 21
term(9) = a+8d = -47
21 + 8d = -47
d = -17/2

using the first sum formula this time
sum(81) = (81/2)(42 + 80(-17/2)) = -25839