Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Write an explicit formula that represents the sequence defined by the following recursive formula: a_(1)=7" and "a_(n)=5a_(n-1) Answer: a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=7 and an=5an−1 a_{1}=7 \text { and } a_{n}=5 a_{n-1} \newlineAnswer: an= a_{n}=

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Convert between explicit and recursive formulasright chevron icon

Full solution

Q. Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=7 and an=5an−1 a_{1}=7 \text { and } a_{n}=5 a_{n-1} \newlineAnswer: an= a_{n}=
  1. Identify Base Case: Identify the base case and the recursive step from the given recursive formula.\newlineThe base case is a1=7a_{1} = 7, which means the first term of the sequence is 77. The recursive step is an=5an−1a_{n} = 5a_{n-1}, which means each term is 55 times the previous term. This indicates that the sequence is geometric with a common ratio of 55.
  2. Write First Few Terms: Write the first few terms of the sequence using the recursive formula to observe the pattern.\newlinea1=7a_{1} = 7\newlinea2=5a1=5×7=35a_{2} = 5a_{1} = 5 \times 7 = 35\newlinea3=5a2=5×35=175a_{3} = 5a_{2} = 5 \times 35 = 175\newlinea4=5a3=5×175=875a_{4} = 5a_{3} = 5 \times 175 = 875\newlineWe can see that each term is 55 times the previous term, confirming that the sequence is geometric.
  3. Derive Explicit Formula: Derive the explicit formula for the geometric sequence.\newlineFor a geometric sequence, the nth term is given by an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}, where a1a_1 is the first term and rr is the common ratio. In this case, a1=7a_1 = 7 and r=5r = 5. Therefore, the explicit formula is an=7⋅5n−1a_n = 7 \cdot 5^{n-1}.

More problems from Convert between explicit and recursive formulas

Question
Find the sum of the finite arithmetic series. ∑n=110(7n+4)\sum_{n=1}^{10} (7n+4)\newline______
Get tutor helpright-arrow

Posted 9 months ago

Question
Type the missing number in this sequence: `55,\59,\63,\text{_____},\71,\75,\79`
Get tutor helpright-arrow

Posted 5 months ago

Question
Type the missing number in this sequence: `2, 4 8`, ______,`32`
Get tutor helpright-arrow

Posted 5 months ago

Question
What kind of sequence is this? 2,10,50,250,…2, 10, 50, 250, \ldots Choices:Choices:\newline[A]arithmetic\text{[A]arithmetic}\newline[B]geometric\text{[B]geometric}\newline[C]both\text{[C]both}\newline[D]neither\text{[D]neither}
Get tutor helpright-arrow

Posted 5 months ago

Question
What is the missing number in this pattern? 1,4,9,16,25,36,49,64,81,____1, 4, 9, 16, 25, 36, 49, 64, 81, \_\_\_\_
Get tutor helpright-arrow

Posted 7 months ago

Question
Classify the series. ∑n=012(n+2)3 \sum_{n = 0}^{12} (n + 2)^3 \newlineChoices:\newline[A]arithmetic\text{[A]arithmetic}\newline[B]geometric\text{[B]geometric}\newline[C]both\text{[C]both}\newline[D]neither\text{[D]neither}
Get tutor helpright-arrow

Posted 5 months ago

Question
Find the first three partial sums of the series.\newline1+6+11+16+21+26+⋯1 + 6 + 11 + 16 + 21 + 26 + \cdots\newlineWrite your answers as integers or fractions in simplest form.\newlineS1=S_1 = ____\newlineS2=S_2 = ____\newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 7 months ago

Question
Find the third partial sum of the series. \newline3+9+15+21+27+33+⋯3 + 9 + 15 + 21 + 27 + 33 + \cdots\newlineWrite your answer as an integer or a fraction in simplest form. \newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 7 months ago

Question
Find the first three partial sums of the series. \newline1+7+13+19+25+31+⋯1 + 7 + 13 + 19 + 25 + 31 + \cdots\newlineWrite your answers as integers or fractions in simplest form. \newlineS1=S_1 = ____\newlineS2=S_2 = ____\newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 9 months ago

Question
Does the infinite geometric series converge or diverge?\newline1+34+916+2764+⋯1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots\newlineChoices:\newline[A]converge\text{[A]converge}\newline[B]diverge\text{[B]diverge}
Get tutor helpright-arrow

Posted 5 months ago

View all (37)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant