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Question Number 195065 by alcohol last updated on 23/Jul/23

	Answered by Rasheed.Sindhi last updated on 23/Jul/23

	$L e t a - d, a, a + d a r e r e q u i r e d n u m b e r s$ $(a - d) + a + (a + d) = 21$ $3 a = 21 \Rightarrow a = 7$ $∴ 7 - d, 7, 7 + d a r e r e q u i r e d n u m b e r s$ $N o w b y g i v e n :$ $(7 - d) + 2, 7 + 3, (7 + d) + 9 a r e i n G P$ $9 - d, 10, 16 + d a r e i n G P$ $\Rightarrow \frac{10}{9 - d} = \frac{16 + d}{10}$ $144 - 16 d + 9 d - d^{2} = 100$ $d^{2} + 7 d - 44 = 0$ $(d + 11) (d - 4) = 0$ $d = - 11, 4$ $∙ d = - 11 :$ $7 - (- 11), 7, 7 + (- 11)$ $18, 7, - 4 (R e j e c t e d b e c a u s e - 4 i s n o t + v e)$ $∙ d = 4 :$ $7 - 4, 7, 7 + 4$ $3, 7, 11 ✓$
	Answered by Rasheed.Sindhi last updated on 23/Jul/23

	$\log (2^{x} + x^{2} - 64) = x (1 - \log 5)$ $\log (2^{x} + x^{2} - 64) = x (\log 10 - \log 5)$ $\log (2^{x} + x^{2} - 64) = x \log (\frac{10}{5})$ $\log (2^{x} + x^{2} - 64) = \log (2^{x})$ $2^{x} + x^{2} - 64 = 2^{x}$ $x^{2} - 64 = 0$ $x = \pm 8$
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