What is an Arithmetic Progression?
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant value is called the Common Difference ($d$).
Staircase & Weekly Savings Model
Imagine saving ₹100 every week. Your weekly savings form an AP: ₹100, ₹200, ₹300, ₹400, ₹500... Each step increases by the exact same height.
Arithmetic Progression is a foundational Modern Math topic that appears directly or indirectly in CAT, XAT, NMAT, SNAP, and MBA CET. It is also used extensively in Algebra word problems, Number Systems, and logical pattern recognition.
Connected Vault Modules:
Use when first term $a$, common difference $d$, and term count $n$ are known.
Fastest sum formula when the last term $l$ is directly given.
Subtract any term from its immediately succeeding term.
Find total terms in a finite AP directly from first and last values.
Finding the 15th Term of an AP
Q: Find the 15th term of the AP: 4, 7, 10, ...
Step 1: Identify first term $a = 4$ and common difference $d = 7 - 4 = 3$.
Step 2: Apply $a_n = a + (n - 1)d$ for $n = 15$:
$$a_{15} = 4 + (15 - 1) \times 3 = 4 + 14 \times 3 = 4 + 42 = \mathbf{46}$$
Sum of a Finite AP Series
Q: Find the sum of the sequence: 5, 10, 15, ..., 50.
Step 1: Identify $a = 5$, last term $l = 50$, and number of terms $n = 10$.
Step 2: Apply shortcut formula $S_n = \frac{n}{2}(a + l)$:
$$S_{10} = \frac{10}{2}(5 + 50) = 5 \times 55 = \mathbf{275}$$
Finding First Term Greater Than Threshold
Q: The first term of an AP is 8 and the common difference is 5. Find the first term greater than 100.
Step 1: Express general term: $a_n = 8 + (n - 1) \times 5$.
Step 2: Set inequality $8 + 5(n - 1) > 100 \implies 5(n - 1) > 92 \implies n - 1 > 18.4 \implies n > 19.4$.
Step 3: Smallest integer $n = 20$.
Step 4: Calculate 20th term: $$a_{20} = 8 + 19 \times 5 = 8 + 95 = \mathbf{103}$$
❌ Trap 1: Confusing $a$ and $d$
Swapping first term & difference
Always double-check that $a$ is the starting term and $d$ is the constant step change!
❌ Trap 2: Using $n$ Instead of $n-1$
a + nd ≠ a_n
The first term requires 0 jumps ($n=1 \implies a + 0d$). The $n$-th term always uses $n - 1$ jumps!
❌ Trap 3: Wrong Term Count in Sum
Guessing $n$ in sum formulas
Never plug values into the sum formula without verifying the exact term count $n$ using $n = [(l - a)/d] + 1$.
⚡ First & Last Term Sum Trick
When first and last terms are known, avoid calculating common difference $d$. Multiply average of first & last term by total terms!
⚡ Quick Term Count Formula
To find how many terms exist between 12 and 97 with step 5: Calculate $(97 - 12) / 5 + 1 = 85 / 5 + 1 = 17 + 1 = \mathbf{18 \text{ terms}}$!
Question 1:
Find the 20th term of the AP: 7, 11, 15, ...
Solution: $a = 7, d = 11 - 7 = 4, n = 20$.
a_{20} = 7 + (20 - 1) × 4 = 7 + 19 × 4 = 7 + 76 = 83.
Question 2:
Find the sum of the first 15 terms of the AP: 3, 6, 9, ...
Solution: $a = 3, d = 3, n = 15$.
S_{15} = (15 / 2) [2(3) + 14(3)] = (15 / 2) [6 + 42] = (15 / 2) × 48 = 15 × 24 = 360.
Question 3:
The sum of the first 25 terms of an AP is 875. If the first term is 11, find the common difference.
Solution: $S_{25} = 875, a = 11, n = 25$.
875 = (25 / 2) [2(11) + 24d] ⇒ 875 × 2 / 25 = 22 + 24d
70 = 22 + 24d ⇒ 24d = 48 ⇒ d = 2.
Q: What is the fundamental definition of an Arithmetic Progression (AP)?
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the Common Difference ($d$).
Q: Can the common difference of an AP be negative?
Yes. If the common difference is negative, the AP is a decreasing sequence (e.g. 20, 15, 10, 5...). If $d = 0$, all terms are equal.
Q: How do I know whether a sequence is an AP?
Check whether the difference between every pair of consecutive terms is identical ($T_2 - T_1 = T_3 - T_2 = ... = d$).