📚 Concepts

📚 Arithmetic Progression (AP):      

A sequence of numbers in which the difference between consecutive terms is constant.

📚 First Term (𝑎):

The starting number of the progression.

📚 Common Difference (𝑑):

The fixed number added to each term to get the next one.      

• (𝑑 = 𝑎₂ − 𝑎₁ = 𝑎₃ − 𝑎₂)

📚 𝑛ᵗʰ Term (𝑇ₙ):      

The term that appears in the 𝑛ᵗʰ position of the sequence.

📚 Sum of First 𝑛 Terms (𝑆ₙ):      

The total of the first 𝑛 numbers in the AP.

📚 Checking an AP:      

If the difference between each term is always the same, it’s an AP.

📚 Types of AP:  

• Increasing AP → 𝑑 > 0  

• Decreasing AP → 𝑑 < 0  

• Constant AP → 𝑑 = 0

✍️ Formulas

✍️ 𝑛ᵗʰ Term of an AP:  

• 𝐓ₙ = 𝐚 + (𝐧 − 1) × 𝐝

✍️ Sum of First 𝑛 Terms (𝐒ₙ):  

• 𝐒ₙ = (𝐧⁄2) × [2𝐚 + (𝐧 − 1) × 𝐝]

• 𝐒ₙ = (𝐧⁄2) × (𝐚 + 𝐥) (where 𝐥 is the last term)

✍️ Common Difference (𝐝):

• 𝐝 = 𝐚₂ − 𝐚₁ = 𝐚₃ − 𝐚₂ = 𝐓ₙ − 𝐓ₙ₋₁

✍️ Finding Number of Terms (𝐧):

• 𝐧 = ((𝐥 − 𝐚) ⁄𝐝) + 1

✍️ Last Term (𝐥) of an AP:

• 𝐥 = 𝐚 + (𝐧 − 1) × 𝐝

✍️ 𝑛ᵗʰ Term from the End:

• 𝐓ₙ (from end) = 𝐥 − (𝐧 − 1) × 𝐝

• 𝐓ₙ = 𝐚 + (𝐧ₜ − 𝐧) × 𝐝 (𝐧ₜ = total number of terms)