The first and the last term of ap is 21 and-47 ,
if the sum of series is 234 calculate ,number of term ,common difference, thesum of first 18 terms
Why was the arithmetic progression so moody? Because it had a lot of ups and downs! Now, let's solve this riddle together.
To find the number of terms, we can use the formula:
n = (last term - first term)/common difference + 1
Substituting the values, we get:
n = (-47 - 21)/d + 1
We can simplify this to:
n = -68/d + 1
Next, let's use the formula for the sum of an arithmetic series:
S = (n/2)(first term + last term)
Substituting the values and simplifying, we get:
234 = (n/2)(21 + (-47))
234 = (n/2)(-26)
Now, let's solve for n:
(n/2)(-26) = 234
-13n = 468
n = -36
Uh-oh, it looks like we've encountered a problem. The number of terms cannot be negative, so there must be an error in the given information. Please recheck the first and last terms of the arithmetic progression.
However, if the values were correct, the common difference (d) would be:
d = (last term - first term)/(n - 1)
d = (-47 - 21)/(-36 - 1)
d = -68/-37
d = 1.84 (approximately)
Finally, to find the sum of the first 18 terms, we can use the formula:
Sn = (n/2)(2a + (n - 1)d)
Substituting the values, we get:
S18 = (18/2)(21 + (18 - 1)1.84)
S18 = 9(21 + 32.72)
S18 = 9(53.72)
S18 ≈ 483.48
Given: The first term (a1) = 21, the last term (an) = -47, the sum of the series (S) = 234.
Step 1: Find the number of terms (n):
an = a1 + (n-1)d
-47 = 21 + (n-1)d
Subtract 21 from both sides:
-47 - 21 = (n-1)d
-68 = (n-1)d
Step 2: Find the common difference (d):
n-1 = -68/d
Step 3: Substituting the values of a1 and an into the formula for the sum of an arithmetic series:
S = (n/2)(a1 + an)
234 = (n/2)(21 + -47)
234 = (n/2)(-26)
Multiply both sides by 2:
468 = -26n
Divide both sides by -26:
n = -468/-26
n = 18
So, the number of terms (n) is 18.
Step 4: Substitute the value of n into the equation from step 2 to find the common difference (d):
n-1 = -68/d
18-1 = -68/d
17 = -68/d
Multiply both sides by d:
17d = -68
Divide both sides by 17:
d = -68/17
d = -4
So, the common difference (d) is -4.
Step 5: Find the sum of the first 18 terms (S18):
S18 = (18/2)(a1 + an)
S18 = (9)(21 + -47)
S18 = (9)(-26)
S18 = -234
Therefore, the sum of the first 18 terms is -234.
To find the number of terms (n), common difference (d), and the sum of the first 18 terms (S18) in an arithmetic progression (AP), we can use the following formulas:
1. Formula for the nth term of an AP (an):
an = a1 + (n - 1)d
2. Formula for the sum of the first n terms of an AP (Sn):
Sn = (n/2)(a1 + an)
Using the given information, we have:
a1 = 21
an = -47
Sn = 234
n = ?
d = ?
S18 = ?
We will first find the common difference (d) using the formula for the nth term of an AP.
1. Substitute the values of a1, an, and n into the formula for the nth term:
an = a1 + (n - 1)d
-47 = 21 + (n - 1)d
2. Simplify the equation:
-68 = (n - 1)d
3. Divide both sides of the equation by (n - 1):
-68 / (n - 1) = d
Now, we can substitute the values of a1, an, and d into the formula for the sum of the first 18 terms to find S18.
4. Substitute the values of a1, an, d, and n=18 into the formula for Sn:
Sn = (n/2)(a1 + an)
234 = (18/2)(21 + (-47))
5. Simplify the equation:
234 = 9(-26)
6. Solve for S18:
S18 = 234/9
S18 = 26
Therefore, the answers to the given problem are:
- Number of terms (n) = 18
- Common difference (d) = -68 / (n - 1)
- Sum of the first 18 terms (S18) = 26