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27 problems match “substitution”. Clear

First term of a recurrence with a constant subtraction ●●○○○ 20 ptsA sequence $a_1, a_2, a_3, \ldots$ satisfies $a_n = 3a_{n-1} - x$ for every $n \ge 2$ , where $x$ is an integer that is the same for all terms. Given that $a_2 = 7$ and $a_4 = 43$ , find $a_1$ .
Function value via substitution ●●○○○ 20 ptsGiven the function $f(x-1)=x^{2}-2x$ , find the value of $f(\sqrt{2})$ .
Two equations with percents ●●○○○ 20 ptsThe sum of $x$ and $y$ is 150. The sum of 5 percent of $x$ and 10 percent of $y$ is 11. Find $x$ and $y$ .
Minimum of a trigonometric sum ●●●○○ 50 ptsFor real numbers $x$ for which all six functions are defined, find the minimum value of $\left|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x\right|$ . Round your answer to two decimal places.
Minimum of a product of trigonometric factors ●●●○○ 50 ptsFind the minimum value of the function $y=(4-3\sin x)(4-3\cos x)$ over real $x$ . Give your answer as a decimal.
Roots in [0,1] after sine substitution ●●●●○ 100 ptsUsing a suitable substitution, determine the number of roots of the equation $8x(1-2x^2)(8x^4-8x^2+1)=1$ that lie in the interval $[0,1]$ .
Exponential equation with a cube root ●●●●○ 100 ptsSolve the equation $27^{x}-7 \sqrt[3]{7 \cdot 3^{x}+6}=6$ .