Physics and Mathematics
Example 1 – Sum of Geometric Progression
Practice Questions with Step-by-Step Solutions
Question 1: Find the sum of the first 6 terms of the G.P. 2, 10, 50, …
Step-by-Step Solution:
First term, [a = 2]
Common ratio, [r = \dfrac{10}{2} = 5]
Number of terms, [n = 6]
Formula for sum of n terms:
[ Sₙ = \dfrac{a(rⁿ − 1)}{r − 1} ]
Substitute values:
[ S₆ = \dfrac{2(5⁶ − 1)}{5 − 1} ]
Calculate power:
[ 5⁶ = 15625 ]
Simplify:
[ S₆ = \dfrac{2(15625 − 1)}{4} ]
Final answer:
[ S₆ = 7812 ]
Question 2: Find the sum of the first 4 terms of the G.P. 7, 14, 28, …
Step-by-Step Solution:
First term, [a = 7]
Common ratio, [r = \dfrac{14}{7} = 2]
Number of terms, [n = 4]
Apply formula:
[ Sₙ = \dfrac{a(rⁿ − 1)}{r − 1} ]
Substitute:
[ S₄ = \dfrac{7(2⁴ − 1)}{1} ]
Calculate power:
[ 2⁴ = 16 ]
Simplify:
[ S₄ = 7(16 − 1) ]
Final answer:
[ S₄ = 105 ]
Question 3: Find the sum of the first 5 terms of the G.P. 9, 3, 1, …
Step-by-Step Solution:
First term, [a = 9]
Common ratio, [r = \dfrac{3}{9} = \dfrac{1}{3}]
Number of terms, [n = 5]
Formula:
[ Sₙ = \dfrac{a(rⁿ − 1)}{r − 1} ]
Substitute values:
[ S₅ = \dfrac{9((\dfrac{1}{3})⁵ − 1)}{\dfrac{1}{3} − 1} ]
Calculate power:
[ (\dfrac{1}{3})⁵ = \dfrac{1}{243} ]
Simplify expression carefully.
Final answer:
[ S₅ = \dfrac{121}{9} ]
Question 4: Find the sum of the first 7 terms of the G.P. 1, −2, 4, …
Step-by-Step Solution:
First term, [a = 1]
Common ratio, [r = −2]
Number of terms, [n = 7]
Formula:
[ Sₙ = \dfrac{a(rⁿ − 1)}{r − 1} ]
Substitute:
[ S₇ = \dfrac{(-2)⁷ − 1}{-2 − 1} ]
Calculate power:
[ (-2)⁷ = -128 ]
Simplify:
[ S₇ = \dfrac{-129}{-3} ]
Final answer:
[ S₇ = 43 ]
Question 5: Find the sum of the first 8 terms of the G.P. 3, 6, 12, …
Step-by-Step Solution:
First term, [a = 3]
Common ratio, [r = 2]
Number of terms, [n = 8]
Formula:
[ Sₙ = \dfrac{a(rⁿ − 1)}{r − 1} ]
Substitute:
[ S₈ = \dfrac{3(2⁸ − 1)}{1} ]
Calculate power:
[ 2⁸ = 256 ]
Simplify:
[ S₈ = 3(256 − 1) ]
Final answer:
[ S₈ = 765 ]
Question 6: Find the sum of the first 6 terms of the G.P. 5, −5, 5, …
Step-by-Step Solution:
First term, [a = 5]
Common ratio, [r = -1]
Number of terms, [n = 6]
Formula:
[ Sₙ = \dfrac{a(rⁿ − 1)}{r − 1} ]
Substitute:
[ S₆ = \dfrac{5((-1)⁶ − 1)}{-1 − 1} ]
Calculate power:
[ (-1)⁶ = 1 ]
Simplify:
[ S₆ = \dfrac{5(1 − 1)}{-2} ]
Final answer:
[ S₆ = 0 ]
Question 7: Find the sum of the first 9 terms of the G.P. 2, 4, 8, …
Step-by-Step Solution:
First term, [a = 2]
Common ratio, [r = 2]
Number of terms, [n = 9]
Apply formula:
[ Sₙ = \dfrac{a(rⁿ − 1)}{r − 1} ]
Substitute:
[ S₉ = \dfrac{2(2⁹ − 1)}{1} ]
Calculate power:
[ 2⁹ = 512 ]
Simplify:
[ S₉ = 2(512 − 1) ]
Final answer:
[ S₉ = 1022 ]
Question 8: Find the sum of the first 5 terms of the G.P. 32, 16, 8, …
Step-by-Step Solution:
First term, [a = 32]
Common ratio, [r = \dfrac{1}{2}]
Number of terms, [n = 5]
Formula:
[ Sₙ = \dfrac{a(rⁿ − 1)}{r − 1} ]
Substitute:
[ S₅ = \dfrac{32((\dfrac{1}{2})⁵ − 1)}{\dfrac{1}{2} − 1} ]
Calculate power:
[ (\dfrac{1}{2})⁵ = \dfrac{1}{32} ]
Simplify carefully.
Final answer:
[ S₅ = 62 ]
Question 9: Find the sum of the first 4 terms of the G.P. 1, 5, 25, …
Step-by-Step Solution:
First term, [a = 1]
Common ratio, [r = 5]
Number of terms, [n = 4]
Apply formula:
[ Sₙ = \dfrac{a(rⁿ − 1)}{r − 1} ]
Substitute:
[ S₄ = \dfrac{5⁴ − 1}{4} ]
Calculate power:
[ 5⁴ = 625 ]
Simplify:
[ S₄ = \dfrac{624}{4} ]
Final answer:
[ S₄ = 156 ]
Question 10: Find the sum of the first 10 terms of the G.P. 3, 9, 27, …
Step-by-Step Solution:
First term, [a = 3]
Common ratio, [r = 3]
Number of terms, [n = 10]
Formula:
[ Sₙ = \dfrac{a(rⁿ − 1)}{r − 1} ]
Substitute:
[ S₁₀ = \dfrac{3(3¹⁰ − 1)}{2} ]
Calculate power:
[ 3¹⁰ = 59049 ]
Simplify:
[ S₁₀ = \dfrac{3(59048)}{2} ]
Final answer:
[ S₁₀ = 88572 ]
Sign in with Google