# Geometric Progression Formulas

## Geometric Progression Formulas

In this Page Geometric Progression Formulas is given that will help in you solving different types of problems.

Geometric Progression
If the ratio of any term to its preceding term is constant in a sequence of numbers then the sequence known as Geometric Progression.

### Formulas for GP

• Common ratio :

Formula for finding the common ratio
$\mathbf{\frac{a_{2}}{a_{1}}}$

• nth term of an GP:

Formula for finding the nth term of an GP $a_{n} = a_{1}r^{n-1}$ where $a_{1}$ = First Term,

r = common ratio and

n = number of Terms

• Sum of first n terms in an GP:
Standard Formula for sum of first n terms in an GP
$\mathbf{\frac{a_{1}(r^{n} -1)}{(r-1)}}$        { if r>1} where, r = common ratio, $a_{1}=$ First Term, n = number of terms
$\mathbf{\frac{a_{1}(1-r^{n}) }{(1-r)}}$        { if r<1} where, r = common ratio, $a_{1}=$ First Term, n = number of terms
• Sum of an infinite GP :

Formula for Sum of an infinite GP

$\mathbf{\frac{a}{1-r}}$                     (if  -1<r<1)

• Geometric Mean (GM) : If two non-zero numbers a and b are in GP, then there GM is GM = $\mathbf{(ab)^{\frac{1}{2}}}$

If three non-zero numbers a,b and c are in GP, then there GM is GM = $\mathbf{(abc)^{\frac{1}{3}}}$

### Properties of Geometric Progression

• If ‘a’ is the first term, r is the common ratio of a finite G.P. consisting of m terms, then the nth term from the end will be =  a rm-n.
• The nth term from the end of the G.P. with the last term ‘l’ and common ratio r is $\mathbf{\frac{l}{r^{n-1}}}$
• Reciprocal of all the term in G.P are also considered in the form of G.P.
• When all terms is GP raised to same power, the new series of geometric progression is form.

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### Using Formulas of Geometric Progression in Questions

Question 1:

What is the 7th term of a geometric progression if the first term (a) is 3 and the common ratio (r) is -2?

A) -192
B) 192
C) -96
D) 96

Solution:
The nth term of a GP is given by an = a * r^(n-1). Substituting the values, a = 3, r = -2, and n = 7:
a7 = 3 * (-2)^(7-1) = 3 * (-2)^6 = 3 * 64 = 192

Question 2:

What is the sum of the first 5 terms of a geometric progression if the first term (a) is 2 and the common ratio (r) is 3/4?

A) 6.75
B) 8.25
C) 7.50
D) 5.50

Solution:
The sum of the first n terms of a GP is given by Sn = a * (r^n – 1) / (r – 1). Substituting the values, a = 2, r = 3/4, and n = 5:
S5 = 2 * ((3/4)^5 – 1) / (3/4 – 1)
= 2 * (243/1024 – 1) / (-1/4)
= 2 * (-781/1024) / (-1/4) = 7.50

Question 3:

Find the common ratio (r) of a geometric progression if the sum of the first 8 terms (S8) is 4374 and the first term (a) is 3.

A) 3
B) 2
C) 4
D) 5

Solution:
The sum of the first n terms of a GP is given by Sn = a * (r^n – 1) / (r – 1). Substituting the values, a = 3, S8 = 4374, and n = 8:
4374 = 3 * (r^8 – 1) / (r – 1)
Now, solving for r:
3 * (r^8 – 1) = 4374 * (r – 1)
r^8 – 1 = 1458 * (r – 1)
r^8 – 1458r + 1457 = 0
Using numerical methods, we find that r ≈ 2.362

Question 4:

What is the 10th term of a geometric progression if the 4th term is 54 and the common ratio (r) is 3?

A) 1458
B) 2187
C) 729
D) None of the above

Solution:
The nth term of a GP is given by an = a * r^(n-1). Substituting the values, a = 54, r = 3, and n = 10:
a10 = 54 * 3^(10-1) = 54 * 3^9 = 54 * 19683 = 1058841

Correct answer: D)None of the above

Question 5:

If the sum of an infinite geometric progression is 72 and the common ratio (r) is 0.5, what is the first term (a)?

A) 2
B) 12
C) 24
D) 36

Solution:
The sum of an infinite GP is given by S = a / (1 – r). Substituting the values, S = 72 and r = 0.5:
72 = a / (1 – 0.5)
72 = a / 0.5
a = 72 * 0.5
a = 36