Question

the terms in the sequence below are divided by 1/2 each time, starting with 1 1/3. Mario thinks that because the terms are being divided, the sequence will eventually have a term that is less than 0. A) Is Mario correct? Write a sentence to explain your answer. B) work out the 4th term in the sequence. Give your answers as a fraction in its lowest terms

Answer 1

A) Mario is incorrect. Despite each term being divided by 1/2, the sequence, starting at 1 1/3, remains positive and approaches but never reaches zero. B) The 4th term is 1/6.

A) Mario's assertion is incorrect. Although each term in the sequence is divided by 1/2, the sequence does not reach or go below zero. This is because the sequence is an infinite geometric progression with a positive initial term (1 1/3). In an infinite geometric sequence, each subsequent term is obtained by multiplying the previous term by a constant factor (in this case, 1/2). As the terms continue to be divided, they approach but never actually reach zero. The sequence exhibits convergence toward zero but remains positive indefinitely.

B) To find the 4th term in the sequence, we use the formula for the nth term of a geometric sequence:
`\(a_n = a_1 * r^((n-1))\), where \(a_1\)`is the first term, \(r\) is the common ratio, and \(n\) is the term number. In this case,
`\(a_1 = 1 (1)/(3)\), \(r = (1)/(2)\), and \(n = 4\)`. Thus, the 4th term is calculated as
`\(a_4 = (1 (1)/(3)) * \left((1)/(2)\right)^3 = (4)/(3)` \times
`(1)/(8) = (1)/(6)\).` Consequently, the 4th term in the sequence is
`\( (1)/(6) \).`