Aimathic
Login | English | Deutsch

Free Math Worksheets

Build your own math worksheets from 21,000 problems for grades 3 to 12, from fractions to calculus. Every problem includes step-by-step solutions.

Sequences as functions

Click problems to add them to your worksheet.

5122318
A sequence is defined by products with alternating factors of \(-2\) and \(-0.5\). Term 1: \(-2\) Term 2: \((-2) \cdot (-0.5)\) Term 3: \((-2) \cdot (-0.5) \cdot (-2)\) Term 4: \((-2) \cdot (-0.5) \cdot (-2) \cdot (-0.5)\) The alternating pattern continues. a) Determine the sign and value of Term 6. b) Determine the sign and value of Term 7. c) What is the absolute value of Term 100? Explain.

Hints

- Calculate the first few terms and look for a repeating pattern. - Track how each new factor changes the preceding term. - Compare terms in even-numbered and odd-numbered positions.

Solution

1. The first four terms are \(-2, 1, -2, 1\). 2. Each pair of factors has product \((-2) \cdot (-0.5) = 1\), so every even-numbered term is \(1\), and every odd-numbered term is \(-2\). 3. Term 6 is positive and equals \(1\). 4. Term 7 is negative and equals \(-2\). 5. Since \(100\) is even, Term 100 is \(1\), so its absolute value is \(1\).

Answer

a) Positive; \(1\) b) Negative; \(-2\) c) \(1\), because every even-numbered term equals \(1\).
5223558
Consider the sequence \(4, 7, 10, 13, \ldots\). a) Write a formula for the \(n\)th term, where \(n=1,2,3,\ldots\). b) Determine whether \(82\) is a term in the sequence. Justify your answer. c) A student claims, “Every term in this sequence has the same remainder when divided by \(3\).” Find the remainder and explain how the formula from part a) shows this.

Hints

- Find the common difference between consecutive terms. - To test whether a number appears, set the term formula equal to that number and solve for \(n\). - Interpret the \(+1\) in \(3n+1\) in relation to multiples of \(3\).

Solution

1. The sequence starts at \(4\) and increases by \(3\), so \(a_n=4+3(n-1)=3n+1\). 2. To test \(82\), solve \(3n+1=82\). Then \(3n=81\), so \(n=27\). Since \(27\) is a positive whole number, \(82\) is the \(27\)th term. 3. The formula \(3n+1\) is one more than a multiple of \(3\), so every term has remainder \(1\) when divided by \(3\).

Answer

a) \(a_n=3n+1\) b) Yes. \(82\) is the \(27\)th term. c) The remainder is \(1\).

All problems may be used, copied and printed free of charge for school and tutoring, including paid tutoring. Commercial adaptations as well as publication or redistribution on the internet are not permitted.

Building free math worksheets