ICSE Class 10 Maths GP Solutions Chapter 11 | ICSE
ICSE Class 10 Maths Geometric Progression Solutions
ICSE Class 10 Maths Chapter 11 Geometric Progression is about sequences in which every term is obtained by multiplying the previous term by the same non-zero number called the common ratio. In Selina Concise Mathematics Class 10, the main skills are finding r, using T_n=ar^{n-1}, finding a finite sum S_n, and using S_\infty only when |r|<1.
What is a geometric progression?
A geometric progression, or G.P., is a sequence of the form a,\ ar,\ ar^2,\ ar^3,\ldots, where a is the first term and r is the common ratio. To test a sequence, divide each term by the term before it. If the ratio remains the same, the sequence is a G.P.
Concept snapshot
Think of a G.P. as one repeated instruction: multiply by r. If r=3, 2 becomes 2,6,18,54,\ldots. If r=-3, the signs alternate: 2,-6,18,-54,\ldots. This is why the sign of r matters in every answer.
Geometric progression formulas for Selina Chapter 11
| Use | Formula | Note |
|---|---|---|
| Common ratio | r=\dfrac{\text{next term}}{\text{previous term}} | The previous term must be non-zero. |
| n^{\text{th}} term | T_n=ar^{n-1} | Use n-1, not n. |
| Finite sum | S_n=\dfrac{a(r^n-1)}{r-1} | Convenient when r>1. |
| Finite sum | S_n=\dfrac{a(1-r^n)}{1-r} | Convenient when |r|<1. |
| Infinite sum | S_\infty=\dfrac{a}{1-r} | Valid only when |r|<1. |
For syllabus confirmation, use the CISCE official website. For supporting algebra notation, use the NCERT textbook portal.
Concise Mathematics Selina Solutions Class 10 ICSE Chapter 11 Geometric Progression: worked solutions
Example 1: First term 8, common ratio -2; find the third term
Step 1: Use T_n=ar^{n-1}.
Step 2: Here a=8, r=-2, and n=3.
T_3=8(-2)^{3-1}=8(-2)^2
T_3=8\times 4=32
Final answer: The third term is 32.
Example 2: Fourth term 16, seventh term 128; find r
Step 1: Let the first term be a and common ratio be r.
T_4=ar^3=16,\qquad T_7=ar^6=128
Step 2: Divide the second equation by the first.
\frac{ar^6}{ar^3}=\frac{128}{16}
r^3=8=2^3
Final answer: r=2.
Example 3: Fifth term 81, second term 24; find the G.P.
Step 1: Convert the terms into equations.
ar^4=81,\qquad ar=24
Step 2: Divide to find r.
\frac{ar^4}{ar}=\frac{81}{24}
r^3=\frac{27}{8}=\left(\frac{3}{2}\right)^3,\qquad r=\frac{3}{2}
Step 3: Find a.
a\left(\frac{3}{2}\right)=24,\qquad a=16
Final answer: The G.P. is 16,\ 24,\ 36,\ 54,\ 81,\ldots.
Example 4: Find 1+3+9+27+\cdots to 12 terms
Step 1: Here a=1, r=3, and n=12.
Step 2: Use S_n=\dfrac{a(r^n-1)}{r-1}.
S_{12}=\frac{1(3^{12}-1)}{3-1}=\frac{531441-1}{2}=265720
Final answer: The sum is 265720.
Example 5: Infinite sum is 9, first term is 3; find r
Step 1: Use S_\infty=\dfrac{a}{1-r}.
9=\frac{3}{1-r}
Step 2: Solve for r.
9(1-r)=3,\qquad 1-r=\frac{1}{3},\qquad r=\frac{2}{3}
Step 3: Check the condition.
\left|\frac{2}{3}\right|<1
Final answer: r=\dfrac{2}{3}, and the infinite sum formula is valid.
Examiner’s mindset for ICSE Class 10 Maths G.P.
Write the formula, substitute values, simplify powers, and state the final answer. In two-term questions, equations like ar^3=16 and ar^6=128 should be divided; do not guess r from the final values. When r is negative, brackets are important because (-2)^2 and -2^2 do not mean the same thing.
Common mistakes in geometric progression
- Using difference instead of ratio: Check \dfrac{\text{next term}}{\text{previous term}}, not subtraction.
- Using ar^n: The correct n^{\text{th}} term is ar^{n-1}.
- Forgetting restrictions: Cancelling x+1 assumes x\neq -1.
- Using infinite sum without checking: S_\infty needs |r|<1.
Related ICSE Class 10 Maths resources
Use the Selina Maths Class 10 solutions index, the ICSE Class 10 Maths resource page, the ICSE Class 10 Maths previous year papers, and ML Aggarwal Class 10 solutions for extra practice.
Frequently Asked Questions
What is the main formula for geometric progression in ICSE Class 10 Maths?
The main formula is T_n=ar^{n-1}, where a is the first term and r is the common ratio.
How do I know whether a sequence is a G.P.?
A sequence is a G.P. if the ratio of each term to the previous term is constant and non-zero.
Which formula is used for the sum of a finite G.P.?
Use S_n=\dfrac{a(r^n-1)}{r-1} or S_n=\dfrac{a(1-r^n)}{1-r}, with r\neq 1. Both are equivalent forms.
When can I use the infinite G.P. formula?
Use S_\infty=\dfrac{a}{1-r} only when |r|<1. If |r|\geq 1, the formula is not valid.