Suppose the sequence a_1, a_2, a_3, ldots is | vedclass

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(A) Let the arithmetic progression be $a_n = a + (n-1)d$,where $a$ is the first term and $d$ is the common difference. Since the terms are distinct,$d

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$2$

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(A) Let the arithmetic progression be $a_n = a + (n-1)d$,where $a$ is the first term and $d$ is the common difference. Since the terms are distinct,$d 
eq 0$.Given that $a_1, a_2, a_4, a_8$ are in geometric progression,let the common ratio be $r$.Thus,$a_1 = a$,$a_2 = ar$,$a_4 = ar^2$,and $a_8 = ar^3$.Substituting the arithmetic progression terms:$a_1 = a$$a_2 = a + d = ar \implies d = a(r-1)$$a_4 = a + 3d = ar^2$$a_8 = a + 7d = ar^3$Substitute $d = a(r-1)$ into the equation for $a_4$:$a + 3(a(r-1)) = ar^2$$a(1 + 3r - 3) = ar^2$$a(3r - 2) = ar^2$Since $a_1, a_2, a_4, a_8$ are in a geometric progression and distinct,$a 
eq 0$,so $3r - 2 = r^2$,which gives $r^2 - 3r + 2 = 0$.Factoring the quadratic equation: $(r-1)(r-2) = 0$.This gives $r = 1$ or $r = 2$.If $r = 1$,then $d = a(1-1) = 0$,which contradicts the condition that the numbers are distinct.Therefore,$r = 2$.

Suppose the sequence $a_1, a_2, a_3, \ldots$ is an arithmetic progression of distinct numbers such that the sequence $a_1, a_2, a_4, a_8, \ldots$ is a geometric progression. The common ratio of the geometric progression is

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