The first and last terms of an a.p are 79 and 5 and it's sum is 814 find(a) the number of terms in the a.p(b)the common different between them.(c) the 9th terms is of an a.p is 52, while the 12th terms is 70 find the sum of it's 20th terms.
What is your school subject? It's not the name of your school.
N (a+l)
/2. A=814,814/2,(79+_5),814/2,84/1=730 or9.6
Let's solve each part of the question step-by-step:
(a) Finding the number of terms (n) in the arithmetic progression (AP):
The formula to find the sum of an arithmetic progression is:
Sum = (n/2) * (first term + last term)
Given:
First term (a₁) = 79
Last term (aₙ) = 5
Sum (S) = 814
Substituting the values in the formula:
814 = (n/2) * (79 + 5)
Now, simplify the equation:
814 = (n/2) * 84
(814 * 2) / 84 = n
19.33 = n
Since the number of terms must be an integer, we take the nearest whole number, which is 19.
Therefore, the number of terms in the arithmetic progression is 19.
(b) Finding the common difference (d) between the terms:
The formula to find the common difference (d) in an arithmetic progression is:
d = (last term - first term) / (number of terms - 1)
Given:
First term (a₁) = 79
Last term (aₙ) = 5
Number of terms (n) = 19
Substituting the values in the formula:
d = (5 - 79) / (19 - 1)
d = -74 / 18
d ≈ -4.11
Therefore, the common difference between the terms is approximately -4.11.
(c) Finding the sum of the 20th terms:
To find the sum of the nth term of an arithmetic progression, we use the formula:
Sum = (n/2) * (first term + last term)
Given:
First term (a₁) = 79
Last term (aₙ) = ?
Number of terms (n) = 20
To find the missing last term, we can use the formula for the nth term of an arithmetic progression:
aₙ = a₁ + (n - 1) * d
Substituting the values:
52 = 79 + (9 - 1) * d
52 = 79 + 8d
8d = 52 - 79
8d = -27
d = -27/8
d ≈ -3.38
Using the common difference (d), we can now find the 12th term:
a₁₂ = 79 + (12 - 1) * d
a₁₂ = 79 + 11 * (-3.38)
a₁₂ = 79 - 37.18
a₁₂ ≈ 41.82
Now we can find the sum of the 20th term using the formula:
Sum = (n/2) * (first term + last term)
Sum = (20/2) * (79 + 41.82)
Sum = 10 * 120.82
Sum = 1208.2
Therefore, the sum of the 20th terms is approximately 1208.2.
To find the number of terms (a) and the common difference (b), we can use the formulas relating to the arithmetic progression (a.p.).
(a) Finding the number of terms:
The sum of an arithmetic progression can be calculated using the formula:
Sum = (number of terms / 2) * (first term + last term)
Given that the sum is 814, the first term is 79, and the last term is 5, we can substitute these values into the formula:
814 = (a / 2) * (79 + 5)
Simplifying the equation:
814 = (a / 2) * 84
814 * 2 = a * 84
1628 = 84a
Now, solving for 'a' (the number of terms) by dividing both sides by 84:
a = 1628 / 84
a ≈ 19.38
Since the number of terms must be a whole number, we round it to the nearest whole number:
a ≈ 19
Therefore, the number of terms in the A.P. is approximately 19.
(b) Finding the common difference:
The common difference (d) can be calculated using the formula:
d = (last term - first term) / (number of terms - 1)
Substituting the given values:
d = (5 - 79) / (19 - 1)
d = -74 / 18
d = -4.11
Rounding it to the nearest whole number gives:
d ≈ -4
Therefore, the common difference (b) of the A.P. is approximately -4.
(c) Finding the sum of the 20th term:
To find the sum of the 20th term, we need to use the formula for the nth term of an arithmetic progression:
Nth term = first term + (n - 1) * common difference
Given that the 9th term is 52, we can substitute these values into the formula:
52 = 79 + (9 - 1) * common difference
52 = 79 + 8 * common difference
Simplifying the equation:
52 = 79 + 8b
Now, let's find the value of b:
52 - 79 = 8b
-27 = 8b
b ≈ -3.375
Rounding it to the nearest whole number gives:
b ≈ -3
Now, let's find the 12th term using the given common difference:
12th term = 79 + (12 - 1) * (-3)
12th term = 79 + 11 * (-3)
12th term = 79 - 33
12th term = 46
Since we know the value of the 12th term is 46, and the common difference is -3, we can use this information to find the sum of the 20th term.
The sum of the arithmetic progression can be calculated using the formula:
Sum = (number of terms / 2) * (first term + last term)
Using the known values:
Number of terms (a) = 20
First term = 79
Last term = 46
Sum = (20 / 2) * (79 + 46)
Sum = 10 * 125
Sum = 1250
Therefore, the sum of the 20th term is 1250.