13Number 13ExplanationThe sequence of real numbers a1,a2,a3,…a_1, a_2, a_3, \dotsa1,a2,a3,… satisfies an+1=4an+1a_{n+1} = 4a_n + 1an+1=4an+1 for natural number nnn and a4=85a_4 = 85a4=85. The value of a3=…a_3 = \dotsa3=…191919202020212121222222232323ExplanationGiven the recursive formula for the sequence: an+1=4an+1a_{n+1} = 4a_n + 1an+1=4an+1 We are asked to find the value of a3a_3a3 given a4=85a_4 = 85a4=85. Use the formula for n=3n = 3n=3: a3+1=4a3+1a_{3+1} = 4a_3 + 1a3+1=4a3+1a4=4a3+1a_4 = 4a_3 + 1a4=4a3+1 Substitute the value a4=85a_4 = 85a4=85: 85=4a3+185 = 4a_3 + 185=4a3+185−1=4a385 - 1 = 4a_385−1=4a384=4a384 = 4a_384=4a3 Divide both sides by 444: a3=844a_3 = \frac{84}{4}a3=484a3=21a_3 = 21a3=21 Thus, the value of a3a_3a3 is 212121.
