Revision: Algebra >> Geometric Progression Maths (English Medium) ICSE Class 10 CISCE

When the numbers (terms) in a sequence are connected to each other by a positive (plus) sign or a negative (minus) sign, the sequence becomes a series. 

A progression is a sequence where each term follows a uniform rule.

• Every progression is a sequence, but with a clear pattern.

A sequence is a group of numbers arranged in a definite order following a rule.

• Numbers in a sequence are called terms or elements.

• The term at position n is called the nth term, denoted by Tₙ.

A sequence, in which each of its terms can be obtained by multiplying or dividing its preceding term by a fixed quantity, is called a geometric progression.

• A fixed number is called the common ratio (r)

r \[=\frac{T_2}{T_1}\]​

Important facts:

• If r > 1→ increasing G.P.

• If 0 < r < 1 → decreasing G.P.

• If r < 0 → terms alternate in sign

nᵗʰ Term from the End \[=\frac{l}{r^{n-1}}\]

When to Use

• G.P. is finite

• Last term, l is known

• Common ratio r is known

T_n ​= arⁿ⁻¹

when you need the nth term (8th term, 15th term, etc.)

l = arⁿ⁻¹

When the G.P. is finite, and you are finding the last term

If r < 1→ use \[S_n=\frac{a(1-r^n)}{1-r}\]

If r > 1 → use \[S_n=\frac{a(r^n-1)}{r-1}\]

If r = 1 → S_n = na

G.M. between a and b

G² = ab 

G =\[\sqrt{ab}\]

G is the geometric mean between a and b.

• Ratio between consecutive terms is constant

• If a, b, c are in G.P. → b² = ac

• Multiplying or dividing all terms by the same non-zero number gives a G.P.

• Reciprocals of terms of a G.P. also form a G.P.

• Corresponding terms of two G.P.s, when multiplied or divided, give another G.P.

Shortcut:
Three terms of a G.P. →\[\frac{a}{r}\], a, ar