Third term of G.P = 6.
Six term of G.P = 48.
Last term of G.P = 3072.
As we know that,
General terms of G.P.
⇒ Tₙ = a.rⁿ⁻¹.
Using this formula in this question, we get.
⇒ T₃ = 6.
⇒ ar³⁻¹ = 6.
⇒ ar² = 6. - - - - - (1).
⇒ T₆ = 48.
⇒ ar⁶⁻¹ = 48.
⇒ ar⁵ = 48. - - - - - (2).
From equation (1) and equation (2), we get.
Divide both the equations, we get.
We get,
⇒ 1/r³ = 1/8.
⇒ r³ = 8.
⇒ r³ = (2)³.
⇒ r = 2.
Put the value of r = 2 in equation (2), we get.
⇒ a(2)⁵ = 48.
⇒ a(32) = 48.
⇒ 2a = 3.
⇒ a = 3/2.
First term : a = 3/2.
Common ratio : r = 2.
To find : Number of term in the G.P.
Put the values in the formula, we get.
⇒ 3072 = (3/2) x (2)ⁿ⁻¹.
⇒ 3072 x 2 = 3 x (2)ⁿ⁻¹.
⇒ 1024 x 2 = (2)ⁿ⁻¹.
⇒ (2)¹⁰ x 2 = (2)ⁿ⁻¹.
⇒ (2)¹¹ = (2)ⁿ⁻¹.
⇒ 11 = n - 1.
⇒ 11 + 1 = n.
⇒ n = 12.
Number of term in the G.P is n = 12.
Option [A] is correct answer.
ᴄᴏʀʀᴇᴄᴛ ᴏᴘᴛɪᴏɴꜱ ᴀʀᴇ ᴀ) ᴀɴᴅ ʙ)
ꜱᴛᴇᴘ -1: ᴡʀɪᴛᴇ ᴍᴀᴛʜᴇᴍᴀᴛɪᴄᴀʟ ᴇQᴜᴀᴛɪᴏɴ ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ Qᴜᴇꜱᴛɪᴏɴ.
ʟᴇᴛ ᴛʜᴇ ꜰɪʀꜱᴛ ᴛᴇʀᴍ ᴏꜰ ᴀ.ᴘ. ʙᴇ ᴀ ᴀɴᴅ ᴄᴏᴍᴍᴏɴ ᴅɪꜰꜰᴇʀᴇɴᴄᴇ ʙᴇ ᴅ,
ꜱᴇᴄᴏɴᴅ ᴛᴇʀᴍ = ᴀ + ᴅ
ᴛʜɪʀᴅ ᴛᴇʀᴍ = ᴀ + 2ᴅ
ꜱɪxᴛʜ ᴛᴇʀᴍ = ᴀ + 5ᴅ
ɢɪᴠᴇɴ, ᴛʜᴀᴛ ᴛʜᴇꜱᴇ ᴀʀᴇ ᴛʜᴇ ᴄᴏɴꜱᴇᴄᴜᴛɪᴠᴇ ᴛᴇʀᴍꜱ ᴏꜰ ᴀ ɢ.ᴘ.
∴ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ ᴏꜰ ɢ.ᴘ., ʀ = ᴀ + ᴅᴀ + 2ᴅ = ᴀ + 2ᴅᴀ + 5ᴅ
ꜱᴛᴇᴘ -2: ꜰɪɴᴅ ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ.
ᴀ + ᴅᴀ + 2ᴅ = ᴀ + 2ᴅᴀ + 5ᴅ
⇒ (ᴀ + 2ᴅ)2 = (ᴀ + ᴅ)(ᴀ + 5ᴅ)
⇒ ᴀ2 + 4ᴀᴅ + 4ᴅ2 = ᴀ2 + 6ᴀᴅ + 5ᴅ2
⇒ ᴅ2 + 2ᴀᴅ = 0
⇒ ᴅ(ᴅ + 2ᴀ) = 0
⇒ ᴅ = 0 ᴏʀ ᴅ = - 2ᴀ
ᴡʜᴇɴ ᴅ = 0, ᴛʜᴇ ᴛᴇʀᴍꜱ ᴏꜰ ɢ.ᴘ. ᴀʀᴇ ᴀ, ᴀ, ᴀ
∴ ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ ɪꜱ 1
ᴡʜᴇɴ ᴅ = - 2ᴀ, ᴛʜᴇ ᴛᴇʀᴍꜱ ᴏꜰ ɢ.ᴘ. ᴀʀᴇ - ᴀ, - 3ᴀ, - 9ᴀ
∴ ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ ɪꜱ 3
ꜰɪɴᴀʟ ᴀɴꜱᴡᴇʀ: ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ ᴏꜰ ɢ.ᴘ ᴄᴀɴ ʙᴇ ᴇɪᴛʜᴇʀ 1 ᴏʀ 3. ᴛʜᴜꜱ, ᴏᴘᴛɪᴏɴꜱ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ᴄᴏʀʀᴇᴄᴛ.
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Answers & Comments
Verified answer
EXPLANATION.
Third term of G.P = 6.
Six term of G.P = 48.
Last term of G.P = 3072.
As we know that,
General terms of G.P.
⇒ Tₙ = a.rⁿ⁻¹.
Using this formula in this question, we get.
Third term of G.P = 6.
⇒ T₃ = 6.
⇒ ar³⁻¹ = 6.
⇒ ar² = 6. - - - - - (1).
Six term of G.P = 48.
⇒ T₆ = 48.
⇒ ar⁶⁻¹ = 48.
⇒ ar⁵ = 48. - - - - - (2).
From equation (1) and equation (2), we get.
Divide both the equations, we get.
⇒ ar² = 6. - - - - - (1).
⇒ ar⁵ = 48. - - - - - (2).
We get,
⇒ 1/r³ = 1/8.
⇒ r³ = 8.
⇒ r³ = (2)³.
⇒ r = 2.
Put the value of r = 2 in equation (2), we get.
⇒ ar⁵ = 48. - - - - - (2).
⇒ a(2)⁵ = 48.
⇒ a(32) = 48.
⇒ 2a = 3.
⇒ a = 3/2.
First term : a = 3/2.
Common ratio : r = 2.
To find : Number of term in the G.P.
⇒ Tₙ = a.rⁿ⁻¹.
Put the values in the formula, we get.
⇒ 3072 = (3/2) x (2)ⁿ⁻¹.
⇒ 3072 x 2 = 3 x (2)ⁿ⁻¹.
⇒ 1024 x 2 = (2)ⁿ⁻¹.
⇒ (2)¹⁰ x 2 = (2)ⁿ⁻¹.
⇒ (2)¹¹ = (2)ⁿ⁻¹.
⇒ 11 = n - 1.
⇒ 11 + 1 = n.
⇒ n = 12.
Number of term in the G.P is n = 12.
Option [A] is correct answer.
ᴄᴏʀʀᴇᴄᴛ ᴏᴘᴛɪᴏɴꜱ ᴀʀᴇ ᴀ) ᴀɴᴅ ʙ)
ꜱᴛᴇᴘ -1: ᴡʀɪᴛᴇ ᴍᴀᴛʜᴇᴍᴀᴛɪᴄᴀʟ ᴇQᴜᴀᴛɪᴏɴ ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ Qᴜᴇꜱᴛɪᴏɴ.
ʟᴇᴛ ᴛʜᴇ ꜰɪʀꜱᴛ ᴛᴇʀᴍ ᴏꜰ ᴀ.ᴘ. ʙᴇ ᴀ ᴀɴᴅ ᴄᴏᴍᴍᴏɴ ᴅɪꜰꜰᴇʀᴇɴᴄᴇ ʙᴇ ᴅ,
ꜱᴇᴄᴏɴᴅ ᴛᴇʀᴍ = ᴀ + ᴅ
ᴛʜɪʀᴅ ᴛᴇʀᴍ = ᴀ + 2ᴅ
ꜱɪxᴛʜ ᴛᴇʀᴍ = ᴀ + 5ᴅ
ɢɪᴠᴇɴ, ᴛʜᴀᴛ ᴛʜᴇꜱᴇ ᴀʀᴇ ᴛʜᴇ ᴄᴏɴꜱᴇᴄᴜᴛɪᴠᴇ ᴛᴇʀᴍꜱ ᴏꜰ ᴀ ɢ.ᴘ.
∴ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ ᴏꜰ ɢ.ᴘ., ʀ = ᴀ + ᴅᴀ + 2ᴅ = ᴀ + 2ᴅᴀ + 5ᴅ
ꜱᴛᴇᴘ -2: ꜰɪɴᴅ ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ.
ᴀ + ᴅᴀ + 2ᴅ = ᴀ + 2ᴅᴀ + 5ᴅ
⇒ (ᴀ + 2ᴅ)2 = (ᴀ + ᴅ)(ᴀ + 5ᴅ)
⇒ ᴀ2 + 4ᴀᴅ + 4ᴅ2 = ᴀ2 + 6ᴀᴅ + 5ᴅ2
⇒ ᴅ2 + 2ᴀᴅ = 0
⇒ ᴅ(ᴅ + 2ᴀ) = 0
⇒ ᴅ = 0 ᴏʀ ᴅ = - 2ᴀ
ᴡʜᴇɴ ᴅ = 0, ᴛʜᴇ ᴛᴇʀᴍꜱ ᴏꜰ ɢ.ᴘ. ᴀʀᴇ ᴀ, ᴀ, ᴀ
∴ ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ ɪꜱ 1
ᴡʜᴇɴ ᴅ = - 2ᴀ, ᴛʜᴇ ᴛᴇʀᴍꜱ ᴏꜰ ɢ.ᴘ. ᴀʀᴇ - ᴀ, - 3ᴀ, - 9ᴀ
∴ ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ ɪꜱ 3
ꜰɪɴᴀʟ ᴀɴꜱᴡᴇʀ: ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ ᴏꜰ ɢ.ᴘ ᴄᴀɴ ʙᴇ ᴇɪᴛʜᴇʀ 1 ᴏʀ 3. ᴛʜᴜꜱ, ᴏᴘᴛɪᴏɴꜱ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ᴄᴏʀʀᴇᴄᴛ.
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