Question

a1=7;an=2an−1+1∀n≥2 find the explicit formula of this

Answer 1

It has given that, a₁ = 7, $a_n=2a_{n-1}+1$ ∀n ≥ 2

To find : The explicit formula of this.

solution : first of all, we should understand, what is the sense of explicit formula.

explicit formula of a sequence is nothing but function in term of n, where n is number of terms.

for instance, $a_n=a+(n-1)d$ is an example of explicit formula.

here, a₁ = 7,

$a_n=2a_{n-1}+1$

for n = 2,

a₂ = 2a₁ + 1 = 2 × 7 + 1

for n = 3,

a₃ = 2a₂ + 1 = 2 × (2 × 7 + 1) + 1 = 2² × 7 + (1 + 2¹)

for n = 4,

a₄ = 2a₃ + 1 = 2 × (2² × 7 + 3) + 1 = 2³ × 7 + (1 + 2¹ + 2²

for n = 5

a₅ = 2a₄ + 1 = 2(2³ × 7 + 7) + 1 = 2⁴ × 7 + (1 + 2¹ + 2² + 2³)

........

for n = n

$a_n=2a_{n-1}+1$ = 2ⁿ¯¹ × 7 + (1 + 2¹ + 2² + 2³ + ..... + 2ⁿ¯²)

= 2ⁿ¯¹ × 7 + 1/2 (2ⁿ - 2)

= 2ⁿ¯¹ × 7 + 2ⁿ¯¹ - 1

= 2ⁿ¯¹ × 8 - 1

Therefore explicit formula of given expression is 2ⁿ¯¹ × 8 - 1