If the sum of the first ten terms of the series is (13/5) + (13/5) + (13/5) + 4 + (13/5) + ..., is (16/5)m, then m is equal to:
Understand the Problem
The question is asking to find the value of 'm' given the sum of the first ten terms of a specific series. The series has terms structured in a particular way, and the challenge is to interpret the series and calculate the required sum to solve for 'm'.
Answer
$m = \frac{31}{4}$
Answer for screen readers
$m = \frac{31}{4}$
Steps to Solve
- Identify the series structure
The series consists of terms where most are $\frac{13}{5}$ and one term is 4. Since the series adds up to 10 terms, we will analyze how many of each term there are.
- Count the occurrences of each term
In the first ten terms:
- There are 7 occurrences of $\frac{13}{5}$ (based on the pattern).
- There is 1 occurrence of 4.
- The remaining 2 terms will also be $\frac{13}{5}$.
- Calculate the total sum of the series
Total Sum = Number of occurrences of $\frac{13}{5}$ × value + occurrence of 4 [ \text{Total Sum} = 8 \times \frac{13}{5} + 4 ] [ = \frac{104}{5} + 4 = \frac{104}{5} + \frac{20}{5} = \frac{124}{5} ]
- Set up the equation for 'm'
We are given that this sum equals $(16/5)m$. Therefore: [ \frac{124}{5} = \frac{16}{5}m ]
- Solve for 'm'
To isolate 'm', multiply both sides by $\frac{5}{16}$: [ m = \frac{124}{5} \times \frac{5}{16} ] [ = \frac{124}{16} = \frac{31}{4} ]
$m = \frac{31}{4}$
More Information
The series mentioned contains a mix of repeating terms and a distinct value. Understanding how often each term occurs is crucial for calculating the sum correctly. The value of 'm' derived represents how the sum correlates to a variable function defined by the series.
Tips
- Forgetting to count all occurrences of each term correctly can lead to an incorrect sum.
- Miscalculating the total sum by not combining the fractions properly can lead to errors in solving for 'm'.
