If each term of a geometric progression with and , is the arithmetic mean of the next two terms and , then equals to

Explanation: \begin{aligned} & \because a_1,a_2,a_3............areinG.P. \\ & & & a_{1}=\frac{1}{8} \\ & & & a_{2}={\frac{a_{3}+a_{4}}{2}}\quad...(\mathbf{i}) \\ & \therefore a_{1}=a (and)\mathrm{^{\prime}r^{\prime}~be~the~com} \mathrm{mon~ratio} \\ & \therefore eq^{n}\left(\mathbf{i}\right)\mathrm{becomes} \\ & & & \mathrm{a}\cdot=\frac{ar^{2}+ar^{3}}{2} \\ & & & \mathrm{2}=r+r^{2} \\ & & r^{2}+r-2 & =0 \\ & & r^{2}+2r-r-2 & =0 \\ & & (r+2)(r-1) & =0 \\ & & & =-2,1 \\ & & & \mathrm{a}\neq a_{1} \\ & \text{一} & & \mathrm{:} & & \neq1 \\ & \mathrm{:} & & =-2 \\ & & \mathrm{Now}\quad\mathrm{S_{20}-S_{18}} & =a_{19}+a_{20} \\ & & & =ar^{18}+ar^{19} \\ & & & =ar^{18}(1+r) \\ & & & =\frac{1}{8}(-2)^{18}(1-2) \\ & & & =\frac{-2^{18}}{23} \end{aligned} -2 15

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JEE 2024 — Mathematics PYQ

JEE | Mathematics | 2024

If each term of a geometric progression $a_{1}, a_{2}, a_{3}, \dots$ with $a_{1} = \frac{1}{8}$ and $a_{2} \neq = a_{1}$ , is the arithmetic mean of the next two terms and $S_{n} = a_{1} + a_{2} + \dots + a_{n}$ , then $S_{20} - S_{18}$ equals to

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