Approximate 1935√

Question

, correct to one decimal place, by using a sequence of values that gets closer and closer to 5–√
.

Answer 1

To approximate $\sqrt{1935}$ using a sequence of values getting closer to $\sqrt{1935}$, we can start by finding an initial approximation for $\sqrt{1935}$ and then refine that approximation.

1. Initial Approximation: First, look for two perfect squares between which 1935 lies:
   • $\sqrt{1936}=44$ (since ${44}^2=1936$)
   • $\sqrt{1935}$ is slightly less than 44.

So, we can start with an initial estimate of $43.9$.

2. Refining the Estimate: We can use the average method (or the method of "successive approximations") to refine the estimate. We use the formula:

$$x_{n+1}=\frac{x_n+\frac{1935}{x_n}}{2}$$

Using $x_0=43.9$:

Calculating $x_1$: $$x_1=\frac{43.9+\frac{1935}{43.9}}{2}$$

Calculating $\frac{1935}{43.9}$: $$\frac{1935}{43.9}\approx 44.1$$ Now, substitute this back into the equation for $x_1$: $$x_1=\frac{43.9+44.1}{2}=\frac{88}{2}=44.0$$

Calculating $x_2$: Now using $x_1=44.0$: $$x_2=\frac{44.0+\frac{1935}{44.0}}{2}$$

Calculating $\frac{1935}{44.0}$: $$\frac{1935}{44.0}\approx 43.9773$$ Substituting this back into the equation for $x_2$: $$x_2=\frac{44.0+43.9773}{2}=\frac{87.9773}{2}\approx 43.98865$$

Calculating $x_3$: Using $x_2\approx 43.98865$: $$x_3=\frac{43.98865+\frac{1935}{43.98865}}{2}$$

Calculating $\frac{1935}{43.98865}$ gives a value close to 44.0, but for practical purposes, we can stop here.

After the sequence of approximations, we find that $\sqrt{1935}\approx 44.0$.

Finally, to correct to one decimal place: $$\sqrt{1935}\approx 44.0$$

Therefore, the approximate value of $\sqrt{1935}$ correct to one decimal place is 44.0.