What is the sum of the 20th term of an ap when the fifth term is 9 and twelfth term is 30
5th term is 9 ------ a + 8d = 9
12 term is 30 ----- a + 11d = 30
subtract them:
3d = 21
d = 7
into a + 8d = 9 gives us a = -47
The 20th term is a + 19d = -47 + 19(7) = 86
sum(20) = 10(first + last) = 10(-47 + 86) = ...
or
= (20/2)( 2(-47) + 19(7) ) = ...
using sum(n) = (n/2)(2a + (n-1)d )
a5 = a1 + 4 d = 9
a12 = a1 + 11 d = 30
Now you must solve system:
a1 + 4 d = 9
a1 + 11 d = 30
The solution is:
a1 = - 2 , d = 3
Sum of the first n terms of an AP::
Sn = n ( a1 + an ) / 2
In this case:
n = 20
a20 = a1 + 19 d = - 3 + 19 * 3
a20 = - 3 + 57
a20 = 54
S20 = 20 ( a1 + a20 ) / 2
S20 = 20 ( - 3 + 54 ) / 2
S20 = 20 * 51 / 2
S20 = 510
To find the sum of the 20th term of an arithmetic progression (AP), we need to know the first term, common difference, and the formula to calculate the sum of terms in an AP.
However, you have provided the values of the 5th term and the 12th term. So, let's start by finding the first term (a) and the common difference (d) of the AP using the given information.
Step 1: Find the common difference (d):
The 5th term (a5) is given as 9.
The 12th term (a12) is given as 30.
We can use these two terms to find the common difference.
d = a12 - a5
d = 30 - 9
d = 21
Step 2: Find the first term (a):
We can calculate the first term (a) using the common difference (d) and the 5th term (a5).
a = a5 - (5 - 1) * d
a = 9 - 4 * 21
a = 9 - 84
a = -75
Now that we know the first term (a = -75) and the common difference (d = 21), we can proceed to find the sum of the 20th term.
Step 3: Find the sum of the 20th term (S20):
The formula to calculate the sum of the first n terms of an arithmetic progression is:
S20 = (n/2)(2a + (n-1)d)
Since we want to find the sum of the 20th term (n = 20), we can substitute the values into the formula.
S20 = (20/2)(2(-75) + (20-1)(21))
S20 = 10(-150 + 19 * 21)
S20 = 10(-150 + 399)
S20 = 10(249)
S20 = 2490
Therefore, the sum of the 20th term of the arithmetic progression is 2490.
To find the sum of the 20th term of an Arithmetic Progression (AP), we need to know the formula for the nth term of an AP.
The formula for the nth term of an AP is given by: 𝑎𝑛 = 𝑎₁ + (𝑛 − 1)𝑑
Where:
𝑎𝑛 is the nth term of the AP,
𝑎₁ is the first term of the AP,
𝑑 is the common difference between the terms of the AP, and
𝑛 is the term number.
Given that the fifth term of the AP is 9, we can substitute those values into the formula to solve for 𝑑.
𝑎₅ = 𝑎₁ + (5 − 1)𝑑
9 = 𝑎₁ + 4𝑑
Similarly, given that the 12th term of the AP is 30, we can substitute those values into the formula to solve for 𝑑 again.
𝑎₁₂ = 𝑎₁ + (12 − 1)𝑑
30 = 𝑎₁ + 11𝑑
Now, we have two equations with two variables (𝑎₁ and 𝑑). We can solve these equations simultaneously to find their values.
From equation 1: 9 = 𝑎₁ + 4𝑑
From equation 2: 30 = 𝑎₁ + 11𝑑
To eliminate 𝑎₁, we can subtract equation 1 from equation 2.
30 - 9 = (𝑎₁ + 11𝑑) - (𝑎₁ + 4𝑑)
21 = 7𝑑
Dividing both sides of the equation by 7 yields: 𝑑 = 3
Now, let's find 𝑎₁ by substituting the value of 𝑑 into equation 1.
9 = 𝑎₁ + 4(3)
9 = 𝑎₁ + 12
𝑎₁ = 9 - 12
𝑎₁ = -3
Now that we have found the values of 𝑎₁ and 𝑑, we can calculate the 20th term of the AP.
𝑎₂₀ = 𝑎₁ + (20 − 1)𝑑
𝑎₂₀ = -3 + 19𝑑
Finally, to find the sum of the 20 terms of the AP, we can use the formula:
Sum = (𝑛/2)(𝑎₁ + 𝑎𝑛)
Sum = (20/2)(-3 + 𝑎₂₀)
Now, substitute the value of 𝑎₂₀ into the formula:
Sum = (20/2)(-3 + 19𝑑)
Sum = 10(-3 + 19𝑑)
Sum = 10(-3 + 19(3))
Sum = 10(-3 + 57)
Sum = 10(54)
Sum = 540
Therefore, the sum of the 20th term of the AP is 540.