Recursive: n th value

We are given two forms of an equation for a sequence: explicit and recursive, and the sequence triples at each step.

Information Given:

• The value of term $n = 6$ is 3645.
• The sequence triples at each step.

Recursive Formula:

The recursive formula can be used to find the next term: $a_n = a_{n-1} \times 3$ Where:

• $a_n$ is the $n$-th term.
• $a_{n-1}$ is the previous term.

We know $a_6 = 3645$, so to find $a_7$ using the recursive formula: $a_7 = a_6 \times 3 = 3645 \times 3 = 10,935$

Thus, the value of the 7th term is $10,935$.

Explicit Formula:

The explicit formula for this geometric sequence is given by: $y = 5 \times 3^x$ To find the 7th term, we substitute $x = 7 - 1 = 6$ (since $x$ represents the position minus 1): $y = 5 \times 3^6 = 5 \times 729 = 3645$ Then for the 7th term: $y = 5 \times 3^7 = 5 \times 2187 = 10,935$

Both methods confirm the 7th term is $10,935$.

Would you like more details or have any further questions?

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Here are 5 related questions for you to explore:

1. What is the 8th term of this sequence using the recursive method?
2. Can you derive the explicit formula for a different geometric sequence?
3. How does the value of the common ratio affect the growth of the sequence?
4. What is the sum of the first 7 terms in this sequence?
5. How can you express this sequence in summation notation?

Tip: For geometric sequences, the ratio between consecutive terms stays constant.