ARITHMETIC PROGRESSIONS
Frequently Asked Questions
A sequence of numbers where the difference between consecutive terms is constant.
The fixed amount added or subtracted to obtain the next term.
Subtract any term from the next: \(d = a_2 - a_1\).
\(a_n = a + (n - 1)d\).
The initial term, denoted by \(a\).
To find any term without listing all previous terms.
\(l = a + (n - 1)d\)
An AP with a fixed number of terms.
An AP that continues indefinitely.
\(S_n = \dfrac{n}{2}\Bigl [2a + (n - 1)d\Bigr]\)
\(S_n = \frac{n}{2} (a + l)\)
Verify if consecutive differences are equal.
Solve \(a + (n - 1)d =\) term and check if n is a positive integer.
The AP grows as n increases.
The AP decreases as n increases.
All terms are equal; constant AP.
4, 7, 10, 13, …
20, 15, 10, 5, …
Yes, if the difference remains constant.
Yes, APs can contain any real numbers.
They help model patterns, growth, and sequences in real life.
Savings plans, seating arrangements, installment payments.
Procedure used to generate the next term: add d each time.
The nth-term formula giving value at any position.
Because \(a_n\) increases linearly with \(n\).
Using wrong values of \(a\) or \(d\), sign errors.
Forgetting parentheses in \(S_n = \dfrac{n}{2}\Bigl [2a + (n - 1)d\Bigr]\).
Value inserted between two numbers to form an AP.
A.M. = \((a + b) / 2\).
Multiple means placed between two numbers by forming a complete AP.
Use \(S_n\) formula and solve quadratic for \(n\).
Using \(l = a + (n - 1)d\) to find unknowns.
The AP still works; terms increase/decrease steadily.
Solve \(a + (n - 1)d = 0\).
Only if \(d = 0\); otherwise terms differ.
Use \(S_n = n/2 (a + l)\) if the last term is known.
\(a_n = a + (n - 1)d\) and \(S_n = \dfrac{n}{2} \Bigl[2a + (n - 1)d\Bigr]\).
Writing AP forward and backward to derive \(S_n\) formula.
Linear increase/decrease by constant steps.
A straight ascending or descending line.
Use the nth-term relation to create equations.
Yes, to find term positions or earlier terms.
Finding term position, sum, or common difference.
Questions involving reasoning, real-life modeling, and pattern analysis.
To solve for \(n\) in sum or \(n\)th-term problems.
In forming equations for sequences and series.
Many patterns in tables or charts show constant increments.
Distance covered in equal intervals increases in AP.
Rearrange nth-term formula: \(a = a_n - (n - 1)d\).
Use \(a_n - a_m = (n - m)d\).
Yes, when deposits increase regularly.
Steps often rise by uniform height increments.
It forms a base for number series, sequences, and reasoning.
A sequence increasing/decreasing in equal increments.
Yes, when the change per period is constant.
Incorrect subtraction for d, choosing wrong \(n\), sign mistakes in equations.
Adjusting AP terms by adding, subtracting, or scaling all values.
New sequence is still an AP with common difference multiplied by that constant.
The resulting sequence remains an AP with unchanged common difference.
Yes, when raises occur in equal annual steps.
