Question

For an arithmetic sequence, a₃ = 67. If the common difference is -6, find:

a. a₁ =
b. the sum of the first 33 terms =
c. a₁ = 55
d. the sum of the first 33 terms = 715

Answer 1

To find the first term, use the formula for the nth term of an arithmetic sequence and solve for a₁. To find the sum of the first 33 terms, use the formula for the sum of an arithmetic series and substitute the values. a₁ = 79, S₃₃ = 2415.

Step-by-step explanation:

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. In this case, the common difference is -6. We are given that a₃ (the third term) is equal to 67. To find a₁ (the first term), we can use the formula:

aₙ = a₁ + (n-1)d

Substituting the given values, we get:

67 = a₁ + 2(-6)

67 = a₁ - 12

a₁ = 67 + 12 = 79

To find the sum of the first 33 terms of the arithmetic sequence, we can use the formula for the sum of an arithmetic series:

Sₙ = (n/2)(a₁ + aₙ)

Substituting the values:

S₃₃ = (33/2)(79 + a₃)

Calculating:

S₃₃ = (33/2)(79 + 67)

S₃₃ = (33/2)(146)

S₃₃ = (33/2)(146) = /highlight>2415