what the recusrsive formula for the sequance of 14 26 48 92 and 180

To find the recursive formula for the sequence \(14, 26, 48, 92, 180\), we need to analyze the pattern and how each term relates to its predecessors.

Given the sequence: \(a_1 = 14, a_2 = 26, a_3 = 48, a_4 = 92, a_5 = 180\).

Let's examine the ratio of successive terms:
\[ \frac{a_2}{a_1} = \frac{26}{14} \approx 1.857 \]
\[ \frac{a_3}{a_2} = \frac{48}{26} \approx 1.846 \]
\[ \frac{a_4}{a_3} = \frac{92}{48} \approx 1.917 \]
\[ \frac{a_5}{a_4} = \frac{180}{92} \approx 1.957 \]

The ratios are not consistent, suggesting the sequence is not geometric. Instead, let's find the differences between successive terms:
\[ a_2 - a_1 = 26 - 14 = 12 \]
\[ a_3 - a_2 = 48 - 26 = 22 \]
\[ a_4 - a_3 = 92 - 48 = 44 \]
\[ a_5 - a_4 = 180 - 92 = 88 \]

We notice that each difference is twice the previous difference, suggesting a pattern involving powers of 2.

Let’s denote the differences as \(d_n\):
\[ d_2 = 12 \]
\[ d_3 = 22 \]
\[ d_4 = 44 \]
\[ d_5 = 88 \]

Here, \(d_n = 2^{n+1}\):

Now we can set up the recursive formula for \(a_n\):

\[ a_{n+1} = a_n + d_n \]
\[ d_{n+1} = 2^{n+1} \]

So, the recursive formula for the sequence is:
\[ a_{n+1} = a_n + 2^{n+1} \]

Let me know if you need further details or have any questions.

Here are 8 related questions that might interest you:
1. How would you derive the explicit formula for this sequence?
2. What is the 10th term of this sequence?
3. How can you prove the correctness of this recursive formula?
4. How does this sequence compare to an arithmetic sequence?
5. Can you find a general solution for any sequence given its recursive formula?
6. How can we modify this sequence to form an arithmetic progression?
7. What are the applications of such sequences in real-world problems?
8. How would you determine if a sequence is converging or diverging?

**Tip:** Understanding the differences between terms in a sequence can often reveal underlying patterns that are not immediately obvious.

Explore the recursive formula for the sequence 14, 26, 48, 92, 180. Learn how to derive the formula step-by-step, understand the pattern recognition involved, and apply it to find any term of the sequence. Suitable for high school students studying sequences and recursive formulas.

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