Question

For each of the following sequences:

1) identify if the sequence is arithmetic, geometric or quadratic . Justify your response.
2) Assuming the first term of each sequence is a₁, give an expression for aᵢ. (In other words, find a formula for the i-th term in the sequence).
3) if the sequence is arithmetic or geometric, compute the sum of the first 15 terms in the sequence. Show your work. If the sequence is quadratic, then show the work used to answer part (b).
(a) 5,15,45,135,…
(b) −8,−3,2,7…
(c) 1,6,12,19…

Answer 1

For the sequence (a): (a) The sequence is geometric. (b) The expression for the i-th term is aᵢ = 5x3^(i-1). (c) The sum of the first 15 terms can be calculated using S = 5(1 - 3^15)/(1 - 3).

Step-by-step explanation:

a) The sequence (a) is a geometric sequence because each term is obtained by multiplying the previous term by the same factor of 3. For example, the second term 15 is obtained by multiplying the first term 5 by 3, and the third term 45 is obtained by multiplying the second term 15 by 3.

b) Assuming the first term is a₁=5, the expression for the i-th term in the sequence is given by aᵢ = 5x3^(i-1).

c) The sum of the first 15 terms in a geometric sequence can be calculated using the formula S = a₁(1 - r^n)/(1 - r), where S is the sum, a₁ is the first term, r is the common ratio, and n is the number of terms. For the sequence (a), the sum is S = 5(1 - 3^15)/(1 - 3).