given-a-ar-ar-2-ar-3-is-a-GPwith-n-r-lt-1-if-a-x-1-x-2-ar-x-3-x-4-x-5-x-6-ar-2-x-7-x-8-x-9-x-10-x-11-x-12-ar-3-where-a-x-1-x-2-ar-AP-ar-x-3-x-4-x-5-x

Question Number 79649 by john santu last updated on 27/Jan/20

given a, ar, {ar}^{2}, {ar}^{3}, \dots is a GPwith

n \to \infty, r < 1

if : a, x_{1}, x_{2}, ar, x_{3}, x_{4}, x_{5}, x_{6}, {ar}^{2},

x_{7}, x_{8}, x_{9}, x_{10}, x_{11}, x_{12}, {ar}^{3}, \dots .

where : a, x_{1}, x_{2}, ar \Rightarrow AP

ar, x_{3}, x_{4}, x_{5}, x_{6}, {ar}^{2} \Rightarrow AP

{ar}^{2}, x_{7}, x_{8}, x_{9}, x_{10}, x_{11}, x_{12}, {ar}^{3} \Rightarrow AP

\dots etc

if \lim_{n \to \infty} (x_{1} + x_{2} + x_{3} + \dots) = \frac{21}{16} \times \frac{a}{1 - r}

what is r ?

Commented by john santu last updated on 27/Jan/20

mister Mjs, W, mind is power i need your help

Commented by mind is power last updated on 27/Jan/20

a, x_{1}, x_{2}, a r A p

s = 2 a (1 + r)

x_{1} + x_{2} = 2 a (1 + r) - a (1 + r) = a (1 + r)

x_{3} + x_{4} + x_{5} + x_{6} = 2 (a r + {a r}^{2})

x_{7} + \dots \dots . . + x_{12} = 3 ({a r}^{2} + {a r}^{4})

w e g e t Σ x_{k} = Σ (a + 2 a r + 3 {a r}^{2} + \dots \dots .) + Σ (a r + 2 {a r}^{2} + \dots \dots . .)

= a Σ (1 + 2 r + 3 r^{2} + \dots .) + a r Σ (1 + 2 r + 3 r^{2} + \dots \dots .)

= (a + a r) Σ (1 + 2 r + 3 r^{2} + \dots .)

\sum_{k ⩾ 1} {k r}^{k - 1} = Σ \frac{d}{d r} (r^{k}) = \frac{d}{d r} Σ r^{k} = \frac{d}{d r} (\frac{r}{1 - r}) = \frac{1}{{(1 - r)}^{2}}

\Rightarrow \frac{a (1 + r)}{{(1 - r)}^{2}} = \frac{21 a}{16 (1 - r)}, r \in] 0, 1 [

\Rightarrow 21 (1 - r) = 16 (1 + r) \Rightarrow 5 = 37 r \Rightarrow r = \frac{5}{37}

Commented by john santu last updated on 27/Jan/20

sir the option

\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} and \frac{1}{6}

Commented by john santu last updated on 27/Jan/20

may be it answer in my book wrong

Commented by john santu last updated on 27/Jan/20

thank you mr Mind is power, W

Commented by mind is power last updated on 27/Jan/20

w i t h e p l e a s u r

Answered by mr W last updated on 27/Jan/20

{a r}^{k}, x_{m + 1}, x_{m + 2}, \dots, x_{m + 2 (k + 1)}, {a r}^{k + 1} a r e A P

w i t h m = \underset{j = 0}{\sum^{k - 1}} 2 (j + 1) = 2 \times \frac{k (k + 1)}{2} = k (k + 1)

s a y t h i s A P i s :

b_{0}, b_{1}, \dots, b_{2 (k + 1)}, b_{2 (k + 1) + 1}

Σ b = \frac{(b_{0} + b_{2 (k + 1) + 1}) (2 (k + 1) + 2)}{2} = (k + 2) ({a r}^{k} + {a r}^{k + 1})

Σ b = {a r}^{k} + Σ x + {a r}^{k + 1} = (k + 2) ({a r}^{k} + {a r}^{k + 1})

\Rightarrow Σ x = (k + 1) ({a r}^{k} + {a r}^{k + 1}) = a (1 + r) (k + 1) r^{k}

\Rightarrow \sum_{a l l} x = \underset{k = 0}{\sum^{\infty}} a (1 + r) (k + 1) r^{k}

\Rightarrow \sum_{a l l} x = \frac{a (1 + r)}{r} \underset{k = 1}{\sum^{\infty}} {k r}^{k}

\Rightarrow \sum_{a l l} x = \frac{a (1 + r)}{r} S

S = 1 r + 2 r^{2} + 3 r^{3} + \dots

r S = 1 r^{2} + 2 r^{3} + 3 r^{4} + \dots

S - r S = r + r^{2} + r^{3} + \dots = \frac{r}{1 - r}

(1 - r) S = \frac{r}{1 - r}

S = \frac{r}{{(1 - r)}^{2}}

\Rightarrow \sum_{a l l} x = \frac{a (1 + r)}{r} \times \frac{r}{{(1 - r)}^{2}} = \frac{a (1 + r)}{{(1 - r)}^{2}}

\Rightarrow \frac{a (1 + r)}{{(1 - r)}^{2}} = \frac{21}{16} \times \frac{a}{1 - r}

\Rightarrow \frac{1 + r}{1 - r} = \frac{21}{16}

\Rightarrow 16 + 16 r = 21 - 21 r

\Rightarrow 37 r = 5

\Rightarrow r = \frac{5}{37}

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