The sum of the first 12 terms of the arithmetic sequence with general term aₙ = 3n+5 is 294.
Step-by-step explanation:
The problem already stated that the sequence we are dealing with is an arithmetic sequence. We just need to know the equation for finding the sum of the first n terms for arithmetic sequence in order to solve the problem.
Given:
n = 12
aₙ = 3n + 5
where n is the number of terms and aₙ is the nth term
Formula for Sum of First n terms of Arithmetic sequence:
S = [2a₁ + (n - 1) (d)]
where S is the sum of the first n terms, a₁ is the first term, and d is the common difference.
We first need to solve for a₁ and d to use the working formula. We can use the general term equation given in the problem which is aₙ = 3n+5 to solve for a₁ and d.
Solving for a₁:
n = 1 (since we are solving for the first term)
aₙ = 3n+5
a₁ = (3)(1) + 5
a₁ = 8
To solve for d, solve for a₂ and a₃ first, then subtract a₂ from a₃, and subtract also a₁ from a₂.
Solving for a₂:
n = 2
aₙ = 3n+5
a₂ = (3)(2) + 5
a₂ = 11
Solving for a₃:
n = 3
aₙ = 3n+5
a₃ = (3)(3) + 5
a₃ = 14
Solving for d:
a₃ - a₂ = 14 - 11 = 3
a₂ - a₁ = 11 - 8 = 3
Since we ended up getting the same answer from subtracting a₂ from a₃, and a₁ from a₂, hence our common difference is 3.
Now that we have the complete values we need, we can now solve the sum using the working equation above.
Solving for the sum of the first 12 terms:
n = 12
S = [2a₁ + (n - 1) (d)]
S = [(2)(8) + (12-1)(3)]
S = 294
Therefore, the sum of the first 12 terms of the arithmetic sequence with general term aₙ = 3n+5 is 294.
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Answer:
The sum of the first 12 terms of the arithmetic sequence with general term aₙ = 3n+5 is 294.
Step-by-step explanation:
The problem already stated that the sequence we are dealing with is an arithmetic sequence. We just need to know the equation for finding the sum of the first n terms for arithmetic sequence in order to solve the problem.
Given:
n = 12
aₙ = 3n + 5
where n is the number of terms and aₙ is the nth term
Formula for Sum of First n terms of Arithmetic sequence:
S = [2a₁ + (n - 1) (d)]
where S is the sum of the first n terms, a₁ is the first term, and d is the common difference.
We first need to solve for a₁ and d to use the working formula. We can use the general term equation given in the problem which is aₙ = 3n+5 to solve for a₁ and d.
Solving for a₁:
n = 1 (since we are solving for the first term)
aₙ = 3n+5
a₁ = (3)(1) + 5
a₁ = 8
To solve for d, solve for a₂ and a₃ first, then subtract a₂ from a₃, and subtract also a₁ from a₂.
Solving for a₂:
n = 2
aₙ = 3n+5
a₂ = (3)(2) + 5
a₂ = 11
Solving for a₃:
n = 3
aₙ = 3n+5
a₃ = (3)(3) + 5
a₃ = 14
Solving for d:
a₃ - a₂ = 14 - 11 = 3
a₂ - a₁ = 11 - 8 = 3
Since we ended up getting the same answer from subtracting a₂ from a₃, and a₁ from a₂, hence our common difference is 3.
Now that we have the complete values we need, we can now solve the sum using the working equation above.
Solving for the sum of the first 12 terms:
n = 12
S = [2a₁ + (n - 1) (d)]
S = [(2)(8) + (12-1)(3)]
S = 294
Therefore, the sum of the first 12 terms of the arithmetic sequence with general term aₙ = 3n+5 is 294.
Step-by-step explanation:
Read more about arithmetic sequence here: brainly.ph/question/825249
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