Question

The sum of first three terms of a G. P is 16 and the sum of the next three terms is 128. Determine the first term, common ratio and the sum to n terms of the G. P..

Answer 1

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ʟᴇᴛ (ａ) ʙᴇ ᴛʜᴇ ғɪʀsᴛ ᴛᴇʀᴍ ᴀɴᴅ ( ｒ ) ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ʀᴀᴛɪᴏ ᴏғ ᴛʜᴇ ɢɪᴠᴇɴ ɢ. ᴘ, ᴛʜᴇɴ

● a + ar + ar² = 16 → a( 1 + r+ r² ) = 16

ᴀɴᴅ

● ar³ + ar⁴ + ${ar}^{5}$ = ar3 ( 1 + r + r² ) = 128

Dividing ( ii ) bye ( i ) , we get

r³ = 8 → r = 2

ᴘᴜᴛᴛɪɴɢ ᴛʜɪs ᴠᴀʟᴜᴇ ᴏғ r (=2 ) in ( i ) , we get

a ( 1 + 2 + 4 ) = 16 → a = 16/7

□ Sn = a( rn - 1 ) / r - 1 = 16 / 7 × 2n - 1 / 2 - 1

= 16 / 7 ( 2n - 1 )