Final answer:
To find the 34th term of the arithmetic sequence 1, 3, 5,..., you can use the formula aₙ = 2n - 1. Substitute n = 34 to find a₃₄ = 67. For the geometric sequence 2, -6, 18,..., you can use the formula aₙ = 2 * (-3)^(n-1). Substitute n = 4 to find a₄ = -54.
Step-by-step explanation:
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. The formula to find the nth term of an arithmetic sequence is given by aₙ = a₁ + (n-1)d, where a₁ is the first term and d is the common difference.
For the first arithmetic sequence, aₙ = 2n - 1. To find the 34th term, substitute n = 34 into the formula:
a₃₄ = 2(34) - 1 = 67
For the second arithmetic sequence, aₙ = -4n - 2. To find the 27th term, substitute n = 27 into the formula:
a₂₇ = -4(27) - 2 = -110
A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant. The formula to find the nth term of a geometric sequence is given by aₙ = a₁ * r^(n-1), where a₁ is the first term and r is the common ratio.
For the third geometric sequence, aₙ = 2 * (-3)^(n-1). To find the 4th term, substitute n = 4 into the formula:
a₄ = 2 * (-3)^(4-1) = -54
For the fourth geometric sequence, aₙ = 4^(n-1). To find the 7th term, substitute n = 7 into the formula:
a₇ = 4^(7-1) = 256