Use iteration to guess an explicit formula for the sequence

Use iteration to guess an explicit formula for the sequence dk = 5d_k1 + 3, for all integers k 1 d0 = 2

Solution

Given d_k = 5d_{k - 1} + 3, ...... (1)

d_{k - 1} = 5d_{k - 2} + 3 ......(2)

Substituting (2) in (1),

d_k = 5(5d_{k - 2} + 3)+ 3

= 5²d_{k - 2} + (5 x 3) + 3 ....... (3)

Now, substituting  d_{k - 2} = 5d_{k - 3} + 3 in (3), we get

d_k = 5²( 5d_{k - 3} + 3) + (5 x 3) + 3

= 5³d_{k - 3} + (5² x 3) + (5 x 3) + 3 ....... (3)

Following the same method,

d_k = 5⁴d_{k - 4} + (5³ x 3) + (5² x 3) + (5 x 3) + 3 ....... (4)

Or, in general,

d_k = 5^rd_{k - r} + [(5^{r - 1} x 3) + (5^{r - 2} x 3) + ...... + (5 x 3) + 3] ....... (5)

The expression within [] in (5) = 3{5^{r - 1 +} 5^{r - 2} + ...... + 5 + 1} = 3{1 + 5 + ..... + 5^{r - 2} + 5^{r - 1}}.... (6)

The expression within {} in (6) is a GP with a(first term) = 1, r(common ratio) = 5 and n(number of terms) = r and hence,

(6) = 3 x 1(5^r - 1)/(5 - 1) = (3/4)(5^r - 1) ....... (7)

Now, substituting (7) in (5), we get,

d_k = 5^rd_{k - r} + (3/4)(5^r - 1) ....... (8)

Putting r = k in (8): d_k = 5^kd₀ + (3/4)(5^k - 1) ....... (9)

Given d₀ = 2, (9) becomes: d_k =2(5^k) + (3/4)(5^k - 1) = (11/4)5^k - (3/4)....... (9)

(9) is the requisite formula.

[To verify if the above formula is correct

Using (1), d₁ = (5 x d₀) + 3 = 13. By (9), d₁ = (11/4)(5) - 3/4 = 13]