The 9th term of an AP is 52 while the 12 term is 70 find the sum of the 20 terms.
answers
https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html
a+8d = 52
a+11d = 70
Now find a and d, and then
S20 = 20/2 (2a+19d)
To find the sum of the 20 terms of an arithmetic progression (AP), we need to determine the common difference (d) and the first term (a₁).
Given:
The 9th term (a₉) = 52
The 12th term (a₁₂) = 70
Step 1: Find the common difference (d)
The formula to find the nth term of an arithmetic progression is:
aₙ = a₁ + (n-1)d
Using the 9th term:
a₉ = a₁ + (9-1)d
52 = a₁ + 8d
Using the 12th term:
a₁₂ = a₁ + (12-1)d
70 = a₁ + 11d
Step 2: Solve the equations simultaneously
Now we have two equations:
52 = a₁ + 8d
70 = a₁ + 11d
To solve these equations, we can subtract the first equation from the second equation:
70 - 52 = (a₁ + 11d) - (a₁ + 8d)
18 = 3d
d = 6
Plugging the value of d back into the first equation, we can solve for a₁:
52 = a₁ + 8(6)
a₁ = 52 - 48
a₁ = 4
Step 3: Find the sum of the 20 terms
The sum of the n terms of an arithmetic progression is given by the formula:
Sₙ = (n/2) * (2a₁ + (n-1)d)
In this case, n = 20, a₁ = 4, and d = 6:
S₂₀ = (20/2) * (2(4) + (20-1)(6))
S₂₀ = 10 * (8 + 19(6))
S₂₀ = 10 * (8 + 114)
S₂₀ = 10 * 122
S₂₀ = 1220
Therefore, the sum of the 20 terms is 1220.