Is it correct to say: Using (n) instead of (n − 1) in the nth-term formula?

a_n = a + (n − 1)d. For a_1 the coefficient of d is 0, not 1.

Is it correct to say: Assuming any progression is an AP?

Verify d = a_2 − a_1 = a_3 − a_2 = a_4 − a_3. Only then it is an AP.

Is it correct to say: Forgetting the factor n/2 in the sum formula?

S_n = (n/2)[2a + (n − 1)d]. Missing the ½ halves the answer.

Is it correct to say: Applying AP formulas to a GP problem?

Check whether difference is constant (AP) or ratio is constant (GP).

Is it correct to say: Wrong sign of d for a decreasing AP?

If terms decrease, d is negative. Example: 20, 17, 14, … has d = −3.

Is it correct to say: Confusing 'nth term' with 'sum to n terms'?

a_n is a single term; S_n is cumulative. Read carefully.

Arithmetic Progression

AP, nth term, sum of n terms, applications — Maharashtra Board Class 10 Algebra

📐 10 formulas✏️ 3 examples🎯 6 practice⚖️ 5-6 marks🏫 Maharashtra State Board📚 Class 10✓ 2025–26 syllabus

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Board Exam Tips

→AP problems are compulsory in the Algebra paper — 4 to 6 marks every year.
→Formula for nth term (a_n = a + (n-1)d) and sum (S_n = n/2 [2a + (n-1)d]) — memorise both forms.
→Word problems on installment savings, salary hikes and prizes are SSC favourites.
→Marathi terms: समांतर श्रेणी (AP), पहिलं पद (first term), सामाईक फरक (common difference).
→State given (a, d, n) clearly at the start — worth 1 identification mark.

📐 Formulas(10)

nth Term of AP★ Board fav

a_{n} = a + (n - 1) d

Symbol	Meaning
$a_{n}$	nth term of the AP
$a$	First term
$d$	Common difference
$n$	Position of the term

Sum of First n Terms★ Board fav

S_{n} = \frac{n}{2} [2 a + (n - 1) d]

Sum in Terms of First and Last

S_{n} = \frac{n}{2} (a + l)

Common Difference from Two Terms

d = a_{n} - a_{n - 1}

General Term from Two Given Terms

d = \frac{a _{p} - a _{q}}{p - q}

Arithmetic Mean

b = \frac{a + c}{2}

Sum of First n Natural Numbers

1 + 2 + 3 + \dots + n = \frac{n ( n + 1 )}{2}

Sum of First n Odd Numbers

1 + 3 + 5 + \dots + (2 n - 1) = n^{2}

Sum of First n Even Numbers

2 + 4 + 6 + \dots + 2 n = n (n + 1)

Term from Sums

a_{n} = S_{n} - S_{n - 1}

✏️ Solved Examples

1Solved Exampleeasy3 steps

Find the 15th term of the AP: 3, 8, 13, 18, …

Identify a and d

a = 3, d = 8 - 3 = 5

2Solved Exampleboard5 steps

The sum of first n terms of an AP is 3n² + 2n. Find the AP and its 20th term.

S_1 gives the first term

S_{1} = 3 (1)^{2} + 2 (1) = 5 \Rightarrow a = 5

3Solved ExampleHOTS5 steps

A man saves Rs 200 in the first month and increases his savings by Rs 25 each month. How much will he save in 2 years? In which month will his savings first exceed Rs 700?

This is an AP with a = 200, d = 25, n = 24

a = 200, d = 25

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Using (n) instead of (n − 1) in the nth-term formula
✓a_n = a + (n − 1)d. For a_1 the coefficient of d is 0, not 1.
2
Assuming any progression is an AP
✓Verify d = a_2 − a_1 = a_3 − a_2 = a_4 − a_3. Only then it is an AP.
3
Forgetting the factor n/2 in the sum formula
✓S_n = (n/2)[2a + (n − 1)d]. Missing the ½ halves the answer.
4
Applying AP formulas to a GP problem
✓Check whether difference is constant (AP) or ratio is constant (GP).
5
Wrong sign of d for a decreasing AP
✓If terms decrease, d is negative. Example: 20, 17, 14, … has d = −3.
6
Confusing 'nth term' with 'sum to n terms'
✓a_n is a single term; S_n is cumulative. Read carefully.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the 10th term of the AP: 2, 7, 12, 17, …
Q2
Find the sum of the first 20 terms of the AP: 1, 4, 7, 10, …
Q3
The 5th term of an AP is 15 and the 12th term is 36. Find the AP.
Q4
How many terms of the AP 9, 17, 25, … are needed to give a sum of 636?
Q5
The sum of three numbers in AP is 27 and their product is 648. Find them.
Q6
Find the sum of all two-digit odd numbers.

📝 Notes

Arithmetic Progression

Maharashtra Board's Algebra paper treats AP as one of the highest-yielding chapters. The formulas are few but must be applied carefully — a single sign or index error costs marks.

SSC angle

2-mark: find a_n given a and d.
3-mark: find sum to n terms or given S_n formula recover the AP.
HOTS: real-life word problem — savings, prizes, salary hikes.

What is an AP?

A sequence in which each term differs from the previous by a fixed number d. Examples:

2, 5, 8, 11, … (d = 3)
100, 90, 80, … (d = −10)
5, 5, 5, 5, … (d = 0 ; constant AP)

Choose your AP form wisely

First term and common difference given → use a_n = a + (n − 1)d.
Last term given → use S_n = (n/2)(a + l).
S_n formula given → a_n = S_n − S_{n-1}.

Handy tricks for word problems

Three numbers in AP: take (a − d), a, (a + d) — sum is 3a.
Four numbers in AP: take (a − 3d), (a − d), (a + d), (a + 3d) — sum is 4a, common difference 2d.
Five in AP: (a − 2d), (a − d), a, (a + d), (a + 2d).

Board vs CBSE

Both boards teach the same material. Maharashtra SSC packs in more real-life word problems (savings, salary hikes, installment plans, prizes) — practise these carefully.