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In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then
(1) the sum of the fifth to eleventh terms (including both) is 2032.
(2) the sum of the sixth and the ninth term is 188.
(3) the first term of the GP. is 2.
(4) the common ratio of the GP. is 1/2.?
(1) the sum of the fifth to eleventh terms (including both) is 2032.
(2) the sum of the sixth and the ninth term is 188.
(3) the first term of the GP. is 2.
(4) the common ratio of the GP. is 1/2.?
Most Upvoted Answer
In a geometric progression the sixth term is eight times the third ter...
Geometric Progression Analysis
To analyze the given geometric progression (GP), we define:
- Let the first term be a.
- Let the common ratio be r.
The nth term of a GP is given by Tn = a * r^(n-1).
Condition 1: Sixth term is eight times the third term
- T6 = a * r^5
- T3 = a * r^2
According to the condition:
T6 = 8 * T3
a * r^5 = 8 * (a * r^2)
=> r^3 = 8
=> r = 2.
Condition 2: Sum of the seventh and eighth terms is 192
- T7 = a * r^6
- T8 = a * r^7
Thus,
T7 + T8 = a * r^6 + a * r^7 = a * r^6 (1 + r) = 192.
Substituting r = 2:
a * 2^6 (1 + 2) = 192
=> a * 64 * 3 = 192
=> a = 1.
Sum of Terms from Fifth to Eleventh
To find the sum from T5 to T11:
Sum = T5 + T6 + T7 + T8 + T9 + T10 + T11
Using known formulas:
= a * r^4 + a * r^5 + a * r^6 + a * r^7 + a * r^8 + a * r^9 + a * r^10
= a * r^4 (1 + r + r^2 + r^3 + r^4 + r^5 + r^6)
= 1 * 2^4 * (1 + 2 + 4 + 8 + 16 + 32 + 64)
= 16 * 127 = 2032.
Verification of Other Conditions
1. Sum of the sixth and ninth terms:
T6 + T9 = a * r^5 + a * r^8 = 1 * 32 + 1 * 256 = 288.
2. First term:
The first term of the GP is indeed 1.
3. Common ratio:
The common ratio is 2, not 1/2.
Conclusion
- The first term is 1.
- The common ratio is 2.
- The sum from T5 to T11 is correctly calculated as 2032.
Thus, statements (1) and the first part of (3) hold true while the common ratio and other conditions do not.
To analyze the given geometric progression (GP), we define:
- Let the first term be a.
- Let the common ratio be r.
The nth term of a GP is given by Tn = a * r^(n-1).
Condition 1: Sixth term is eight times the third term
- T6 = a * r^5
- T3 = a * r^2
According to the condition:
T6 = 8 * T3
a * r^5 = 8 * (a * r^2)
=> r^3 = 8
=> r = 2.
Condition 2: Sum of the seventh and eighth terms is 192
- T7 = a * r^6
- T8 = a * r^7
Thus,
T7 + T8 = a * r^6 + a * r^7 = a * r^6 (1 + r) = 192.
Substituting r = 2:
a * 2^6 (1 + 2) = 192
=> a * 64 * 3 = 192
=> a = 1.
Sum of Terms from Fifth to Eleventh
To find the sum from T5 to T11:
Sum = T5 + T6 + T7 + T8 + T9 + T10 + T11
Using known formulas:
= a * r^4 + a * r^5 + a * r^6 + a * r^7 + a * r^8 + a * r^9 + a * r^10
= a * r^4 (1 + r + r^2 + r^3 + r^4 + r^5 + r^6)
= 1 * 2^4 * (1 + 2 + 4 + 8 + 16 + 32 + 64)
= 16 * 127 = 2032.
Verification of Other Conditions
1. Sum of the sixth and ninth terms:
T6 + T9 = a * r^5 + a * r^8 = 1 * 32 + 1 * 256 = 288.
2. First term:
The first term of the GP is indeed 1.
3. Common ratio:
The common ratio is 2, not 1/2.
Conclusion
- The first term is 1.
- The common ratio is 2.
- The sum from T5 to T11 is correctly calculated as 2032.
Thus, statements (1) and the first part of (3) hold true while the common ratio and other conditions do not.
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In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then(1) the sum of the fifth to eleventh terms (including both) is 2032.(2) the sum of the sixth and the ninth term is 188.(3) the first term of the GP. is 2.(4) the common ratio of the GP. is 1/2.?
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In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then(1) the sum of the fifth to eleventh terms (including both) is 2032.(2) the sum of the sixth and the ninth term is 188.(3) the first term of the GP. is 2.(4) the common ratio of the GP. is 1/2.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then(1) the sum of the fifth to eleventh terms (including both) is 2032.(2) the sum of the sixth and the ninth term is 188.(3) the first term of the GP. is 2.(4) the common ratio of the GP. is 1/2.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then(1) the sum of the fifth to eleventh terms (including both) is 2032.(2) the sum of the sixth and the ninth term is 188.(3) the first term of the GP. is 2.(4) the common ratio of the GP. is 1/2.?.
In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then(1) the sum of the fifth to eleventh terms (including both) is 2032.(2) the sum of the sixth and the ninth term is 188.(3) the first term of the GP. is 2.(4) the common ratio of the GP. is 1/2.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then(1) the sum of the fifth to eleventh terms (including both) is 2032.(2) the sum of the sixth and the ninth term is 188.(3) the first term of the GP. is 2.(4) the common ratio of the GP. is 1/2.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then(1) the sum of the fifth to eleventh terms (including both) is 2032.(2) the sum of the sixth and the ninth term is 188.(3) the first term of the GP. is 2.(4) the common ratio of the GP. is 1/2.?.
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In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then(1) the sum of the fifth to eleventh terms (including both) is 2032.(2) the sum of the sixth and the ninth term is 188.(3) the first term of the GP. is 2.(4) the common ratio of the GP. is 1/2.?, a detailed solution for In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then(1) the sum of the fifth to eleventh terms (including both) is 2032.(2) the sum of the sixth and the ninth term is 188.(3) the first term of the GP. is 2.(4) the common ratio of the GP. is 1/2.? has been provided alongside types of In a geometric progression the sixth term is eight times the third term and sum of the seventh and the eight term is 192, then(1) the sum of the fifth to eleventh terms (including both) is 2032.(2) the sum of the sixth and the ninth term is 188.(3) the first term of the GP. is 2.(4) the common ratio of the GP. is 1/2.? theory, EduRev gives you an
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