Common Terms in Arithmetic Progression: JEE Main & Advanced Guide

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AP Basics: Formulas You Need Before Starting 

Before working with common terms in arithmetic progressions, these standard AP formulas are essential.

The nth term of an AP:

an=a+(n−1)⋅da_n = a + (n-1) \cdot dan​=a+(n−1)⋅d

Where:

• a = first term
• d = common difference
• n = term number

Sum of n terms of an AP:

Sn=n2[2a+(n−1)⋅d]=n2(a+l)S_n = \frac{n}{2} \left[2a + (n-1) \cdot d\right] = \frac{n}{2}(a + l)Sn​=2n​[2a+(n−1)⋅d]=2n​(a+l)

Where l is the last term.

These are the only two formulas you will need throughout this entire topic. The common terms concept is built entirely on top of them.

💡 Key Reminder: The number of terms in any AP is always a positive integer. If your calculation gives a decimal value for n, always take the integer part (floor value) — never round up.

For a solid foundation in AP from Class 10 NCERT, refer to NCERT Solutions Class 10 Maths. The Class 11 extension of this topic is covered in NCERT Solutions Class 11 Maths.

What Are Common Terms in Two APs? 

Consider two arithmetic progressions — call them AP₁ and AP₂. Each has its own first term and its own common difference. Some terms will appear in both APs. These shared values are called the common terms of the two APs.

The key property to know:

The common terms of two APs always form another AP.

This new AP — call it the Common AP — has the following properties:

This result extends naturally to three APs. The common terms of three APs also form an AP, where the common difference is the LCM of d₁, d₂, and d₃.

How to Find the First Common Term Between Two APs 

Finding the first common term requires observation. Write out both APs term by term until you spot a value that appears in both sequences.

Example:

• AP₁: 1, 5, 9, 13, 17, 21 ... (first term = 1, common difference = 4)
• AP₂: 2, 7, 12, 17, 22 ... (first term = 2, common difference = 5)

Scanning both, the value 17 appears in AP₁ and also in AP₂. That is the first common term. So the Common AP starts at 17.

Its common difference = LCM(4, 5) = 20.

Therefore, the Common AP is: 17, 37, 57, 77 ...

For small APs, writing terms by hand is fine. For larger APs with many terms, spotting the first common term by observation is still the right method — once identified, the LCM handles the rest automatically.

How to Find the Number of Common Terms in Two APs 

Once you have the Common AP (first term and common difference), follow these steps:

1. Find the last term of AP₁ and the last term of AP₂ using the nth term formula.
2. Take the smaller of the two last-term values. This is the upper limit for the Common AP.
3. Set up the inequality: the nth term of the Common AP ≤ smaller last-term value.
4. Solve for n. If n is not an integer, take the floor value (integer part only).

This floor value is your number of common terms.

💡 Why use the smaller last term? A common term must exist in both APs. If a value is in AP₁ but exceeds the last term of AP₂, it cannot be common. So the effective range is limited by whichever AP ends earlier — in terms of value, not count.

How to Find the Sum of Common Terms in Two APs 

Once you know:

• Number of common terms (n)
• First common term (a)
• Common difference of the Common AP (d = LCM of d₁ and d₂)

Apply the standard sum formula directly:

Sn=n2[2a+(n−1)⋅d]S_n = \frac{n}{2}\left[2a + (n-1) \cdot d\right]Sn​=2n​[2a+(n−1)⋅d]

No new formula is needed. The Common AP behaves exactly like any other AP.

Worked Example 1: Finding Number of Common Terms (JEE Main Level) 

Question: Find the number of common terms in the AP 4, 9, 14, … up to 25 terms, and the AP 3, 6, 9, 12, … up to 37 terms.

Step 1: Identify both APs.

• AP₁: first term a₁ = 4, common difference d₁ = 5, number of terms = 25
• AP₂: first term a₂ = 3, common difference d₂ = 3, number of terms = 37

Step 2: Find the last terms.

a25=4+24×5=4+120=124a_{25} = 4 + 24 \times 5 = 4 + 120 = 124a25​=4+24×5=4+120=124 a37=3+36×3=3+108=111a_{37} = 3 + 36 \times 3 = 3 + 108 = 111a37​=3+36×3=3+108=111

Step 3: Find the first common term.

Writing both APs: 4, 9, 14, 19, 24 ... and 3, 6, 9, 12, 15 ...

The value 9 appears in both. First common term = 9.

Step 4: Find the common difference of the Common AP.

$$\text{LCM}(5,3)=15$$

Common AP: 9, 24, 39, 54, 69, 84, 99 ...

Step 5: Find the number of common terms.

Smaller last term = 111 (from AP₂).

Set up: $9+(n-1)\times 15\le 111$

$(n-1)\times 15\le 102$

$n-1\le 6.8$

$n\le 7.8$

Since n must be an integer: n = 7.

Answer: 7 common terms.

Worked Example 2: Directly Given Last Term Values (JEE Main Level)

Question: Find the number of terms common to the two APs: 3, 7, 11, … up to 407, and 2, 9, 16, … up to 709.

Here the last term values (407 and 709) are given directly — no need to calculate them.

Step 1: Identify both APs.

• AP₁: a₁ = 3, d₁ = 4, last term = 407
• AP₂: a₂ = 2, d₂ = 7, last term = 709

Step 2: Find the first common term.

Writing terms: AP₁ gives 3, 7, 11, 15, 19, 23 ... AP₂ gives 2, 9, 16, 23 ...

The value 23 appears in both. First common term = 23.

Step 3: Common difference of Common AP.

$$\text{LCM}(4,7)=28$$

Common AP: 23, 51, 79 ...

Step 4: Number of common terms.

Smaller last term = 407 (from AP₁).

Set up: $23+(n-1)\times 28\le 407$

$(n-1)\times 28\le 384$

$n-1\le \mathrm{13.7...}$

$n\le \mathrm{14.7...}$

Answer: n = 14 common terms.

Worked Example 3: Three APs and JEE Advanced 2019 Question 

This question appeared in JEE Advanced 2019. It demonstrates that the same concept extends directly from two APs to three.

Question (paraphrased from transcript): Three infinite APs are given:

• AP₁: 1, 4, 7, 10, ... (d₁ = 3)
• AP₂: 2, 7, 12, 17, 22, ... (d₂ = 5)
• AP₃: 3, 10, 17, 24, ... (d₃ = 7)

Find the first term (A) and common difference (D) of the AP formed by terms common to all three. Then find A + D.

Step 1: Common difference of the Common AP.

$$D=\text{LCM}(3,5,7)=105$$

Step 2: Find the first common term.

For three APs, writing all three term by term and finding the first triple overlap is the approach. A smart shortcut: since AP₂ has d₂ = 5, its unit digits cycle between 2 and 7 only (adding 5 repeatedly). So any common term must end in 2 or 7.

Now scan AP₃ (which has the largest common difference, so fewest terms to check per range): 3, 10, 17, 24, 31, 38, 45, 52 ...

52 ends in 2 — it qualifies as a candidate. Check whether 52 is in AP₁:

$$52=1+(n-1)\times 3\Rightarrow 51=(n-1)\times 3\Rightarrow n=18$$

18 is an integer. So 52 is the 18th term of AP₁. ✓

52 is also in AP₂ (ends in 2, consistent with AP₂'s pattern). ✓

52 is also in AP₃ (we found it there directly). ✓

First common term A = 52.

Step 3: Answer.

$$A+D=52+105=157$$

💡 Faculty insight: "This question appeared in JEE Advanced 2019. It looks difficult — but when you read it carefully, the concept is exactly the same: find the first common term, take LCM of common differences, done. Questions that look dangerous in the paper are often the easiest ones if you know the concept."

For further practice with Class 12 Maths exam questions, refer to the Class 12 Maths Question Paper 2023. For Class 10 AP foundation, see Class 10 Maths Question Paper 2023.

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