In-a-geometric-sequence-of-real-numbers-the-sum-of-the-first-two-terms-is-7-and-the-sum-of-the-first-six-terms-is-91-The-sum-of-the-first-four-terms-is-

Question Number 111542 by Aina Samuel Temidayo last updated on 04/Sep/20

In a geometric sequence of real

numbers, the sum of the first

two terms is 7 and the sum of the first

six terms is 91 . The sum of the first

four terms is .

Commented by bemath last updated on 04/Sep/20

w h a t t h e m e a n i n g s u m o f t w o t w o t e r m s ?

Commented by Aina Samuel Temidayo last updated on 04/Sep/20

I meant the sum of the first two terms .

Answered by Lordose last updated on 04/Sep/20

S_{2} = \frac{a (1 - r^{2})}{1 - r} = a (1 + r) = 7 \Rightarrow (1)

S_{6} = \frac{a (1 - r^{6})}{1 - r} = a (1 + r) (r^{4} + r^{2} + 1) = 91 \Rightarrow (2)

\frac{(2)}{(1)} = r^{4} + r^{2} + 1 = 13

set r^{2} = x

x^{2} + x - 12 = 0

x^{2} + 4 x - 3 x - 12 = 0

x (x + 4) - 3 (x + 4) = 0

x = 3 or x = - 4

r = \sqrt{3} or r = 2 i

Since r {can}^{'} t be complex, r = \sqrt{3}

Sub in (1)

a (1 + \sqrt{3}) = 7

a = \frac{7}{1 + \sqrt{3}}

S_{4} = \frac{a (1 - r^{4})}{1 - r} = \frac{7 (1 - {(\sqrt{3})}^{4})}{(1 + \sqrt{3}) (1 - \sqrt{3})}

S_{4} = \frac{7 (- 8)}{- 2} = 28

★ LorD OsE

Commented by Aina Samuel Temidayo last updated on 04/Sep/20

Thanks .

Answered by Rasheed.Sindhi last updated on 04/Sep/20

a + a r = 7

a + a r + {a r}^{2} + {a r}^{3} + {a r}^{4} + {a r}^{5} = 91

a + a r + {a r}^{2} + {a r}^{3} = ?

a = \frac{7}{1 + r}

a + a r + r^{2} (a + a r + {a r}^{2} + {a r}^{3}) = 91

a + a r + {a r}^{2} + {a r}^{3} = \frac{91 - (a + a r)}{r^{2}}

= \frac{91 - 7}{r^{2}} = \frac{84}{r^{2}}

a + a r + {a r}^{2} + {a r}^{3} + {a r}^{4} + {a r}^{5} = 91

a (1 + r + r^{2} + \dots + r^{5}) = 91

(\frac{7}{1 + r}) (1 + r + r^{2} + \dots + r^{5}) = 91

(\frac{1}{1 + r}) (1 + r + r^{2} + \dots + r^{5}) = 13

\frac{1 - r^{6}}{1 - r^{2}} = 13

\frac{(1 - r^{2}) (1 + r^{2} + r^{4})}{1 - r^{2}} = 13

r^{4} + r^{2} - 12 = 0

(r^{2} + 4) (r^{2} - 3) = 0

\underset{r i s n o t r e a l}{r^{2} = - 4} ∣ r^{2} = 3

S u m o f f i r s t f o u r t e r m s = \frac{84}{r^{2}}

= \frac{84}{3} = 28

(c o r r e c t e d)

Commented by Aina Samuel Temidayo last updated on 04/Sep/20

Seen .