A broken calculator can double any number or permute its digits (but 0 cannot go up front). Can we start with number 1 and obtain 78 by a series of these operations

it's impossible to obtain 78

Step-by-step explanation: Since the calculator can double any number or permute its digits except 0 

starting with number 1 

1, 2, 4, 8, 16, 32, 64, 128

Therefore, it's impossible to obtain 78 by a series of these operations

10010010001010010101001001010010100101010010101010010010101010110010001011010010010001

step-by-step explanation:

(a) {a₁, a₁r, a₁r², … a₁rⁿ⁻¹, …}

(b) $59 398

step-by-step explanation:

(a) the annual sequence of profits

aₙ = a₁rⁿ⁻¹

the model for the sequence of profits is {a₁, a₁r, a₁r², … a₁rⁿ⁻¹, …} .

(b) profit after 15 yr

  a₁ = $ 30 000

  r = 1.05

a₁₅ = 30 000 × 1.05¹⁴

a₁₅ = 30 000 × 1.979 93

a₁₅ = $59 398

the profit in year 15 will be $59 398.