Question

If \(\mathrm{t}_{\mathrm{n}}=3^{\mathrm{n}-1}\), then \(\mathrm{S}_{6}-\mathrm{S}_{5}=\) ________. (1) 243 (2) 81 (3) 77 (4) 27

Short answer

In conclusion, the difference between the sums \(S_6\) and \(S_5\) in the given sequence \(t_n = 3^{n-1}\) is 243. We found this by identifying the terms not common in both sums and computing the value of the 6th term \(t_6 = 243\), which was the difference we needed.

Step-by-step solution

Identify the terms in the sums

We are given the sequence defined by \(t_n = 3^{n-1}\). To calculate the sums \(S_6\) and \(S_5\), we need to identify the terms in these sums. The sum \(S_6\) will include the first 6 terms and \(S_5\) will include the first 5 terms of the given sequence.

Calculate the 6th term

As mentioned in the Analysis, the difference between the sums \(S_6\) and \(S_5\) can be found by subtracting the terms in the sum that is not common in both sums, which is the 6th term \(t_6\) in this problem. Now, let's find the value of \(t_6\) using the formula: \(t_6 = 3^{6-1} = 3^5 = 243\)

Compare with given options

Now, compare the value of the 6th term that we computed in Step 2 with the given options: 1) 243 2) 81 3) 77 4) 27 As we see that \(t_6 = 243\), which matches with Option (1), so the correct answer is \(\boxed{243}\). Therefore, the difference between the sums \(S_6\) and \(S_5\) is 243.

Key concepts

Series and Sums in Arithmetic Sequences

In arithmetic sequences, series and sums play a crucial role in understanding patterns and trends. Let's break down this concept step by step. A sequence is essentially a list of numbers following a certain rule. An arithmetic sequence is when each number, or term, is obtained by adding a fixed number to the previous term. The sum of a sequence, often called a "series," sums up all terms in the sequence up to a certain point.

In the given exercise, we look at the sums labeled as \(S_6\) and \(S_5\). Here, \(S_6\) refers to the sum of the first 6 terms of the sequence, while \(S_5\) refers to the sum of the first 5 terms. To find \(S_6 - S_5\), we effectively focus on the additional term included in \(S_6\) but not in \(S_5\), which is exactly the 6th term itself. This tells us that often in such exercises, the key to understanding lies in identifying and focusing on the unique terms that make up the difference.

Pattern Recognition in Sequences

Pattern recognition is a fundamental skill in mathematics, particularly when working with sequences. Recognizing how each term builds on a previous one allows us to predict future terms easily. This pattern can be identified by closely examining the rule provided for generating the sequence.

In the context of the exercise, the sequence follows the formula \(t_n = 3^{n-1}\). With this guideline, each term can be generated by raising 3 to the power of \(n-1\). Therefore, identifying the given pattern becomes essential to solving the problem, as the specific rule informs us how terms change as the sequence progresses. Recognizing these patterns early on in problem-solving can simplify what seems like a complex issue at first glance.

Exponentiation in Arithmetic Sequences

Exponentiation is a mathematical operation involving numbers that are raised to the power of another number, known as the exponent. It signifies repetitive multiplication of a base number. In arithmetic sequences where terms are calculated using an exponent, understanding this concept is necessary.

In this exercise, the term formula \(t_n = 3^{n-1}\) incorporates exponentiation. Here, 3 is the base, while \(n-1\) is the exponent, showing how each term in the sequence is generated by raising 3 to one less than its position number. Applying exponent laws correctly is crucial when working with such sequences.

• Recognize the base and exponent when given \(t_n\).
• Remember that exponentiation allows quick computation of large numbers, like \(3^5 = 243\), without manually multiplying the base several times.
• Applying this helps solve the problem by swiftly calculating specific terms.