The exponential function y(x)=Ceαx satisfies the conditions y(0)=2 and y(1)=1. (a) Find the constants C and α. Enter the exact value of α. Enclose arguments of functions, numerators, and denominators in parentheses. For example, sin(2x) or (a−b)/(1+n).

a = -0.6931471806

C = 2

Step-by-step explanation:

Given that y(x) = Ce^(ax) satisfies the conditions y(0) = 2, and y(1) = 1.

We need to find the values of a and C.

To do this, we are going to apply the conditions given to the function y(x), this way, we will obtain two equations which can then be solved simultaneously.

Applying y(0) = 2, put y(x) = 2 and x = 0 in y(x) = Ce^(ax)

2 = Ce^(a × 0)

C = 2 (because e^0 = 1)(1)

Again, applying y(1) = 1, put y = 1 and x = 1 in y(x) = Ce^(ax)

1 = Ce^(a × 1)

Ce^a = 1 (2)

From (1), C = 2. Putting this in (2), we have

2e^a = 1

Divide both sides by 2

e^a = 1/2

This equation has an index which could make finding a prove difficult by a direct mean, to find a, we need to take logarithm of both sides,

log(e^a) = log(1/2)

(a)log(e) = log(1/2)

a = log(1/2)/(log e)

a = (-0.3010299957)/(0.4342944819)

a = -0.6931471806

see explanation

step-by-step explanation:

since the sequence is geometric there is a common ratio r between consecutive terms, that is

r = 4 ÷ - 2 = - 2

r = - 8 ÷ 4 = - 2

r = 16 ÷ 8 = - 2

the next term - 13 must be incorrect as it should be - 32 for the sequence to be geometric.

the first 4 terms are therefore geometric with common ratio r = - 2