Arithmetic series and inverse problems
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| common difference/ˈkɒmən ˈdɪfrəns/ | công sai |
Can a stated total tell us exactly how many terms were added?
- Two known terms reveal an arithmetic sequence.
- This lesson studies common difference 公差: The fixed amount added to each term to obtain the next.
The method
- 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.
Solving 3n²+5n-350=0 for a term count gives n=10 and n=-35/3. Why keep only 10?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The number of terms is a count: positive and whole. n=-35/3 fails both, and rounding it would give a total that does not satisfy the sum formula - check S_10=175 directly.
A worked case
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Subtract the two term equations to get 4d=12, then a=10−2×3=4.
Where it goes wrong
- 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.
Not true: A noninteger solution for an exact term count can be rounded to the nearest integer. A number of terms must be a whole number; a noninteger solution means no term takes that value.
Find its common difference d.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
d=(22−10)/(7−3)=3.
A noninteger solution for an exact term count can be rounded to the nearest integer.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A number of terms must be a whole number; a noninteger solution means no term takes that value.
Applying it
- 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.
How many terms give total 175 in this sequence?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The sum equation factors to (n−10)(3n+35)=0; retain the positive integer 10.
Match each arithmetic-series formula to its quantity.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The nth term holds n-1 differences, not n; inverse problems solve simultaneous equations for a and d before any sum is touched.
In your course
- AQA A-level Mathematics (7357), specification section D.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Common difference: The fixed amount added to each term to obtain the next.
- For first term a and common difference d, u_n=a+(n−1)d.
- Watch for: the nth term and sum of n terms are different quantities.