Understanding the Series
Before calculating the sum of the series, let's analyze the given terms: 3 1/2, 7, 10 1/2, 14.
- The first term is 3.5
- The second term is 7
- The third term is 10.5
- The fourth term is 14
Identifying the Pattern
To find the pattern in the series, observe the differences between consecutive terms:
- Difference between 7 and 3.5: 3.5
- Difference between 10.5 and 7: 3.5
- Difference between 14 and 10.5: 3.5
This indicates a consistent increase of 3.5 for each subsequent term.
Finding the General Term
We can express the n-th term of the series (Tn) as follows:
Tn = 3.5 + (n-1) * 3.5, where n starts from 1.
This simplifies to Tn = 3.5n.
Calculating Up to the 17th Term
For the 17th term:
T17 = 3.5 * 17 = 59.5.
Summing the Series
To find the sum of the first 17 terms, we use the formula for the sum of an arithmetic series:
Sum = n/2 * (first term + last term).
- Here, n = 17,
- First term = 3.5,
- Last term = 59.5.
Thus, the sum becomes:
Sum = 17/2 * (3.5 + 59.5) = 17/2 * 63 = 17 * 31.5 = 535.5.
Final Result
The sum of the series up to the 17th term is 535.5.