Arithmetic series and inverse problems
| English | 中文 | Pinyin |
|---|---|---|
| common difference/ˈkɒmən ˈdɪfrəns/ | 公差 | gōng chāi |
Two known terms reveal an arithmetic sequence. Can a stated total tell us exactly how many terms were added?
- Two known terms reveal an arithmetic sequence. Can a stated total tell us exactly how many terms were added?
- This lesson studies common difference 公差: The fixed amount added to each term to obtain the next.
Choose the mathematical structure
- For first term a and common difference d, u_n=a+(n−1)d. The sum of n terms is S_n=n[2a+(n−1)d]/2=n(a+u_n)/2. Pairing the first and last terms gives the same pair total throughout. An inverse problem may require solving simultaneous equations for a,d or a quadratic for n; a term count must be a positive integer.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines common difference?
The fixed amount added to each term to obtain the next.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Given u₃=10 and u₇=22, write a+2d=10 and a+6d=22. Subtraction gives d=3 and a=4, so u_n=3n+1 and S_n=n(3n+5)/2. To find a total of 175, solve 3n²+5n−350=0=(n−10)(3n+35). The roots are 10 and −35/3; only n=10 is a positive integer. Check u₁₀=31 and S₁₀=10(4+31)/2=175. A total of exactly 200 has no integer n: S₁₀=175 and S₁₁=209, and these positive terms make the sums strictly increasing.
Arithmetic series and inverse problems
For first term a and common difference d, u_n=a+(n−1)d
Check which series formula answers the question and whether its conditions hold.
Find a when u₃=10 and u₇=22 in an arithmetic sequence.
Subtract the two term equations to get 4d=12, then a=10−2×3=4.
Test a tempting shortcut
- The nth term and sum of n terms are different quantities. The nth term contains n−1 differences, not n. Do not round a noninteger solution for an exact number of terms. If the question asks for the first total at least a target, check the neighbouring integer totals. Not every arithmetic series has increasing partial sums.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A noninteger solution for an exact term count can always be rounded to the nearest integer. This claim is false. Explain which definition or assumption it violates.
Find its common difference d.
d=(22−10)/(7−3)=3.
A noninteger solution for an exact term count can always be rounded to the nearest integer.
The nth term and sum of n terms are different quantities. The nth term contains n−1 differences, not n. Do not round a noninteger solution for an exact number of terms. If the question asks for the first total at least a target, check the neighbouring integer totals. Not every arithmetic series has increasing partial sums.
Interpret a new situation
- For this sequence, the least n giving S_n≥200 is 11, since the previous total is 175. State whether a problem asks for an exact total or a threshold. The sum formula still works for negative common differences, but any positivity or monotonicity assumption must be checked from the actual terms.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
How many terms give total 175 in this sequence?
The sum equation factors to (n−10)(3n+35)=0; retain the positive integer 10.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · D. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The fixed amount added to each term to obtain the next. Choose the relationship, show the method, check its assumptions and interpret the result.