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@AMANDEEP KAUR SEKHON  a1, a2, a3, …. is an A.P. and b1, b2, b3, …… is a G.P. Then the sequence  is said to be an arithmetic-geometric progression. The sum of infinite term is
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use the concept    Arithmetic mean of two numbers (AM) - - wherein It is to be noted that the sequence a, A, b, is in AP where, a and b are the two numbers.       Geometric mean of two numbers (GM) - - wherein It is to be noted that a,G,b are in GP and a,b are two non - zero numbers.     Harmonic mean (HM) of two numbers a and b - -     multiply and divide by...
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As we have learned

Sum of n terms of a GP -

$S_{n}= \left\{\begin{matrix} a\frac{\left ( r^{n}-1 \right )}{r-1}, &if \: r\neq 1 \\ n\, a, & if \, r= 1 \end{matrix}\right.$

- wherein

$a\rightarrow$ first term

$r\rightarrow$ common ratio

$n\rightarrow$ number of terms

$S_{20}= 1 +\left ( \frac{4-1}{2} \right )+\left ( \frac{8-1}{4} \right )+\left ( \frac{16-1}{8} \right )+......$

$=1+2+2.......20times....\left ( 1/2+1/4+1/8+......20times \right )$

$39-1/2(\left ( \frac{1-(1/2)^{19}}{1-1/2} \right ))= 39-1+1/(2)^{19}$

$38+\frac{1}{2^{19}}$

Option 1)

$38+\frac{1}{2^{19}}$

This is correct

Option 2)

$38+\frac{1}{2^{20}}$

This is incorrect

Option 3)

$39+\frac{1}{2^{20}}$

This is incorrect

Option 4)

$39+\frac{1}{2^{19}}$

This is incorrect

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As we learnt in  General term of a GP -   - wherein first term common ratio    Let first term is a andcommon ratio is r then Also   put in Option 1) 7290 Incorrect Option Option 2) 320 correct Option Option 3) 640 Incorrect Option Option 4) 2430 Incorrect Option
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As we learnt in Sum of n terms of an AP - and Sum of n terms of an AP - wherein first term common difference number of terms   Arithmetic mean of n numbers -   - wherein are the n numbers    Let the AP is given   Now                    Series is   upto terms mean Option 1)  26.5 Incorrect option Option 2)  28 Incorrect option Option 3) 29.5 Correct option Option...
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As we learnt in

Common ratio of a GP (r) -

The ratio of two consecutive terms of a GP

- wherein

eg: in 2, 4, 8, 16, - - - - - - -

r = 2

and in 100, 10, 1, 1/10 - - - - - - -

r = 1/10

${A}, {B},{C}$ are in  G.P where ${A}, {B},{C}$  are Terms of an A.P

Let first term is a and common difference is d, and common ratio be r then

A = a+d

B = a+4d

C= a+8d

$\therefore \: \frac{B}{A}=\frac{C}{B}=\frac{r}{1}$

$\frac{a+4d}{a+d} =\frac{a+8d}{a+4d}=\frac{r}{1}$

$\therefore \frac{a+4d+a+d}{a+4d-a-d}=\frac{r+1}{r-1}$

$\Rightarrow \frac{2a+5d}{3d}=\frac{r+1}{r-1}-----(i)$

$\frac{a+8d+a+4d}{a+8d-a-4d} =\frac{r+1}{r-1}$

$\Rightarrow \frac{2a+12d}{4d}=\frac{r+1}{r-1}------(ii)$

from (i) and (ii)

$\frac{2a+5d}{2a+12d}=\frac{3}{4}$

$8a+20d=6a+36d$

2a=16d

a=8d

$\Rightarrow \frac{r+1}{r-1}=\frac{2\times8d+5d}{3d}$$=\frac{16d+5d}{3d}$$=\frac{21}{3}=7$

$\therefore r+1=7r-7$

$8=6r$

$r=\frac{4}{3}$

Option 1)

Incorrect option

Option 2)

Correct option

Option 3)

Incorrect option

Option 4)

Incorrect option

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