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Binomial Theorem Part 2

Section: Mathematics

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Question : 2 of 100

Marks: +1, -0

If each term of a geometric progression

a_1, a_2, a_3, \ldots

a_1 = \frac{1}{8}

a_2 \neq a_1

, is the arithmetic mean of the next two terms and

S_n=a_1+a_2+\cdots+a_n

S_{20} - S_{18}

[29-Jan-2024 Shift 2]

$2^{15}$
$-2^{18}$
$2^{18}$
$-2^{15}$

Solution: 👈: Video Solution

Let

r'

th term of the GP be

a r^{n-1}

. Given,

2 a_r = a_{r+1} + a_{r+2}

2 a r^{n-1} = a r^n + a r^{n+1}

\frac{2}{r} = 1 + r

r^2 + r - 2 = 0

Hence, we get,

r=-2(

as

r \neq 1)

So,

S_{20} - S_{18} =

(Sum upto 20 terms) - (Sum upto

18 \text{ terms } ) = T_{19} + T_{20}

T_{19} + T_{20} = a r^{18}(1 + r)

Putting the values

a = \frac{1}{8}

and

r = -2

;

we get

T_{19}+T_{20}=-2^{15}

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