The Question
Given the arithmetic series: 5 + 7 + 9 + … + 93. Determine the general term of the series, T_n, in the form T_n = pn + q.
Details
💡 Hint
What’s the starting number and the consistent jump between numbers in this sequence? Use these to build your general formula!
📝 Solution Steps
- Identify the first term (a) of the series.
- Calculate the common difference (d) between consecutive terms.
- Substitute the values of ‘a’ and ‘d’ into the general formula for the nth term of an arithmetic sequence: T_n = a + (n-1)d.
- Simplify the expression to the form T_n = pn + q.
📚 Explanation
An arithmetic series is characterized by a constant difference between consecutive terms. First, identify the first term (a) of the series. Then, calculate the common difference (d) by subtracting any term from its succeeding term (e.g., 7 – 5 = 2). Once you have ‘a’ and ‘d’, substitute these values into the general formula for the nth term of an arithmetic sequence: T_n = a + (n-1)d. Finally, simplify the expression to match the required form T_n = pn + q.
✅ Answer
T_n = 2n + 3
⚠️ Common Mistakes
- Incorrectly calculating the common difference.
- Algebraic errors when simplifying the formula after substitution.
📐 Formulas Required
- T_n = a + (n-1)d
