Chapter Overview
🧭Your visual roadmap through the chapter
Sequences
Ordered lists of terms; arithmetic vs geometric sequences
Geometric Progression
Common ratio, nth term, sum of n terms
Infinite Geometric Series
Sum to infinity when the common ratio is less than 1
Numerical sequences formed by a definite law — arithmetic progression (constant difference) and geometric progression (constant ratio) — and how to find any term or the sum of a series.
Progressions model recurring business situations — salary increments, loan installments, savings plans, and growth at a constant rate — and underpin later financial mathematics topics.
Identify AP vs GP; apply the nth term and sum formulas; solve for an unknown term count (n) including via the quadratic formula; evaluate sum to infinity.
Key Definitions
📘Source-based terminology, exam-ready
Sequence
A succession of terms arranged in a definite order and formed according to a definite law, i.e. the set {u1, u2, u3, ... un, ...}.
Key word: definite order
Series
The expression u1+u2+u3+...+un, i.e. the sum of all terms of a sequence; symbolically expressed as Σun.
Key word: sum of terms
Term
Each successive number forming a sequence; denoted T1, T2, ... Tn where Tn is the nth or "general" term.
Key word: Tn = general term
Arithmetic Progression (A.P.)
A series in which the successive terms increase or decrease by a constant common difference (d).
Key word: constant difference
Common Difference (d)
The constant amount obtained by subtracting any term of an A.P. from the term immediately following it.
Key word: d = subtract
Geometric Progression (G.P.)
A sequence where the ratio between a term and the one immediately preceding it is constant throughout; neither the first term nor the ratio can be zero.
Key word: constant ratio
Common Ratio (r)
The constant ratio linking consecutive terms of a G.P.; found by dividing a term by the term preceding it.
Key word: r = divide
Infinite Series
A sequence with no last term. As n approaches infinity, rⁿ approaches zero if |r| < 1, allowing a finite sum to be found.
Key word: no last term
Core Concept
🎯Explained in exam-oriented language
CONCEPT → Arithmetic Progression
MEANING
Each term is linked to the previous term by adding/subtracting a constant (d).
PURPOSE
Model quantities that change by a fixed amount each period.
APPLICATION
Salary increments, fixed-step savings plans, loan installments rising by a flat amount.
CONCEPT → Geometric Progression
MEANING
Each term is linked to the previous term by multiplying by a constant (r).
PURPOSE
Model quantities that grow or shrink by a fixed percentage each period.
APPLICATION
Interest earned, depreciation over time, installments rising by a fixed %, infinite-series totals.
Key Rules & Principles
📏Every source rule, exam-ready
RULE 1 nth term relationship (A.P.)
The coefficient of d is always one less than the term number
2nd term = a+d, 3rd term = a+2d, ... nth term = a+(n−1)d
RULE 4 Properties of an A.P. under transformation
Add/subtract a constant k to every term → still an A.P., same common difference d
Multiply every term by k → still an A.P., new common difference = d × k
Divide every term by k → still an A.P., new common difference = d ÷ k
RULE 2 Choosing the sum formula (A.P.)
Use S = n/2 (a+l) when the first term AND last term are known
Use S = n/2 [2a+(n−1)d] when the first term AND common difference are known
With any 3 of the 5 quantities (a, d, l, n, S) known, the other 2 can be found
RULE 5 Choosing the sum formula (G.P.)
If r < 1, use S = a(1−rⁿ)÷(1−r) — more convenient
If r > 1, use S = a(rⁿ−1)÷(r−1) — more convenient
If r = 1, every term equals a, so S = n × a
RULE 3 Representing unknown A.P./G.P. terms
A.P. 3 terms: (a−d), a, (a+d) | 4 terms: (a−3d),(a−d),(a+d),(a+3d)
G.P. 3 terms: a/r, a, ar | 4 terms: a/r³, a/r, ar, ar³
RULE 6 Sum to infinity — existence condition
Exists ONLY if |r| < 1 (r between −1 and 1)
As n→∞, rⁿ→0, so the rⁿ term can be ignored
If r ≥ 1, the sum grows without limit — no finite sum to infinity
Formula Zone
🧮Every formula supported by the source text
A.P. — NTH TERM
a = first term, d = common difference, n = term number. Tn is also called "l" (the last term) when it's the final term of a finite series.
A.P. — SUM OF N TERMS
First form needs first term (a) & last term (l); second form needs first term (a) & common difference (d).
ARITHMETIC MEAN (A.M.)
The single value A that sits between two quantities a and b so that a, A, b form an A.P.
G.P. — NTH TERM
a = first term, r = common ratio, n = term number. Neither a nor r can be zero.
G.P. — SUM OF N TERMS
Both formulas are mathematically equivalent; pick whichever avoids a negative denominator for easier calculation.
SUM TO INFINITY OF A G.P.
Only valid when |r| < 1. a = first term, r = common ratio.
Representing Unknown Terms (for solving simultaneous problems)
A.P. (3 terms): (a−d), a, (a+d) A.P. (5 terms): (a−2d),(a−d),a,(a+d),(a+2d)
G.P. (3 terms): a/r, a, ar G.P. (5 terms): a/r², a/r, a, ar, ar²
Calculation Roadmap
🗺️General path for any progression question
1· Read Question
Identify given terms/values
2· Identify Type
Constant difference (A.P.) or constant ratio (G.P.)?
3· Select Formula
nth term, sum, or sum to infinity
4· Substitute
Put known values into the formula
5· Calculate & Check
Solve, discard invalid roots, state answer
Step-by-step Methods
🪜Kept separate — do not mix methods
METHOD A · Find the nth Term of an A.P.
- 1Identify the first term (a) and common difference (d)
- 2Identify which term number (n) is required
- 3Substitute into Tn = a + (n−1)d
- 4Calculate and state the final term value
METHOD D · Find the nth Term & Sum of a G.P.
- 1Identify the first term (a) and find r by dividing a term by the one before it
- 2Substitute into Tn = arⁿ⁻¹ for the required term
- 3Choose the sum formula matching whether r<1 or r>1
- 4Substitute and calculate the sum S
METHOD B · Find the Sum of an A.P.
- 1Identify a, and either d or the last term (l)
- 2Choose the matching sum formula (Rule 2)
- 3Substitute the known values
- 4Calculate the sum S
METHOD E · Sum to Infinity of a G.P.
- 1Find the common ratio r
- 2Check |r| < 1 — if not, sum to infinity does not exist
- 3Substitute a and r into S∞ = a ÷ (1−r)
- 4Calculate the final sum
METHOD C · Solve for n (Unknown Number of Terms)
- 1Substitute known a, d and S into S=n/2[2a+(n−1)d]
- 2Multiply out and rearrange into standard quadratic form
- 3Solve using the quadratic formula
- 4Discard the negative/invalid root; keep the sensible n
Reading a Sequence (visual)
A.P. example 2, 6, 10, 14 → each term is +4 from the last (add a constant). G.P. example 16, 8, 4, 2 → each term is ×½ of the last (multiply by a constant).
Different Cases / Scenarios
🔀Recognise the pattern, pick the right route
IF each term differs by a fixed amount added/subtracted
→ Sequence is an A.P. — use nth term / sum formulas
IF each term differs by a fixed multiplying factor
→ Sequence is a G.P. — use ratio-based formulas
IF first term (a) and last term (l) are both known
→ USE S = n/2 (a+l)
IF first term (a) and common difference (d) are known
→ USE S = n/2 [2a+(n−1)d]
IF solving for n produces a quadratic equation
→ Solve normally, then discard the negative/nonsensical root
IF the G.P. common ratio |r| < 1
→ Sum to infinity exists: S∞=a÷(1−r)
IF the G.P. common ratio |r| ≥ 1
→ Sum to infinity does not exist (grows without limit)
IF asked to find 3/4/5 unknown terms of a progression
→ Represent symmetrically around a: e.g. A.P. (a−d),a,(a+d)
Comparison Tables
⚖️Don't confuse these!
| Arithmetic Progression (A.P.) | Geometric Progression (G.P.) | |
|---|---|---|
| Link between terms | Add/subtract a constant (d) | Multiply by a constant (r) |
| nth term | Tn = a + (n−1)d | Tn = arⁿ⁻¹ |
| Sum of n terms | S = n/2(a+l) or n/2[2a+(n−1)d] | S = a(1−rⁿ)/(1−r) or a(rⁿ−1)/(r−1) |
| Sum to infinity | Not applicable | Exists only if |r| < 1: S∞=a/(1−r) |
| G.P. Sum — Case | Condition | Result |
|---|---|---|
| Case I | r = 1 | Every term equals a; S = n × a |
| Case II | r < 1 | rⁿ shrinks toward 0 as n grows; S = a(1−rⁿ)/(1−r), and S∞=a/(1−r) exists |
| Case III | r > 1 | rⁿ grows without bound; S = a(rⁿ−1)/(r−1), and S∞ is infinite (does not exist) |
Properties of an A.P. under transformation (recap)
| Transformation applied to every term | New common difference |
|---|---|
| Add or subtract a constant k | Same as original (d) |
| Multiply by a constant k | d × k |
| Divide by a constant k | d ÷ k |
Worked Calculation Logic
✍️Method students must reproduce in the exam
GIVEN → FORMULA → SUBSTITUTION → CALCULATION → ANSWER
A company funds an 18-month expansion: Rs. 100,000 in month 1, increasing Rs. 10,000 per month. Find the 18th payment and the total invested.
GIVEN: a = 100,000, d = 10,000, n = 18 (A.P.)
FORMULA: Tn = a+(n−1)d then S = n/2[2a+(n−1)d]
SUBSTITUTION: T18 = 100,000+(17)(10,000) | S = 18/2[2(100,000)+(17)(10,000)]
CALCULATION: T18 = 100,000+170,000 = 270,000 | S = 9[200,000+170,000] = 9(370,000)
FINAL ANSWER: 18th payment = Rs. 270,000; total invested = Rs. 3,330,000
Exam Question Patterns
📝Identification — is a given sequence an A.P. or a G.P.?
nth term — direct formula application
Sum calculation — choosing the correct sum formula
Solve-for-n — rearranging into a quadratic equation
Sum to infinity — existence check then calculation
Business application word problems — salary, loans, savings, depreciation
Exam Traps
⚠️⚠ WATCH OUT
- ✗ Misidentifying A.P. vs G.P. — check whether terms ADD or MULTIPLY by a constant.
- ✗ Wrong sum formula — using n/2(a+l) when l is unknown, instead of the d-based formula.
- ✗ Keeping a negative n — the number of terms must always be positive; discard the invalid root.
- ✗ Applying sum to infinity when |r| ≥ 1 — it simply does not exist in that case.
- ✗ Common ratio sign errors — dividing terms in the wrong order or mishandling a negative ratio.
Quick Decision Tree
🌳Chapter 3 in 60 Seconds
⚡A.P. nth term: a+(n−1)d
A.P. sum: n/2(a+l) or n/2[2a+(n−1)d]
G.P. nth term: arⁿ⁻¹
G.P. sum: a(1−rⁿ)/(1−r) [r<1]
Sum to infinity: a/(1−r), only if |r|<1
A.M.: (a+b)/2 Solve for n → quadratic, discard negative root
Trap: A.P. adds, G.P. multiplies
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