Terms of the Series:
The given series is √2, 2√2, 4√2, ...

Aim:
To find the number of terms required to reach the value 14 + 7√2.

Approach:
We can observe that each term in the series is obtained by multiplying the previous term by 2.

Step 1: Find the nth term of the series
Let the nth term of the series be Tn. We can write the nth term as Tn = 2^(n-1) * √2.

Step 2: Solve the equation
We need to find the value of n for which Tn = 14 + 7√2.
Substituting the values in the equation, we get:
2^(n-1) * √2 = 14 + 7√2

Step 3: Simplify the equation
To simplify the equation, we can divide both sides by √2.
2^(n-1) = (14 + 7√2) / √2
2^(n-1) = 7√2 + 7
2^(n-1) = 7(√2 + 1)

Step 4: Take the logarithm of both sides
To isolate n, we can take the logarithm of both sides of the equation.
log(2^(n-1)) = log(7(√2 + 1))
(n-1)log(2) = log(7) + log(√2 + 1)

Step 5: Solve for n
Now we can solve for n by dividing both sides of the equation by log(2).
n-1 = (log(7) + log(√2 + 1)) / log(2)
n = 1 + (log(7) + log(√2 + 1)) / log(2)

Step 6: Calculate the value of n
Using a calculator to evaluate the right-hand side of the equation, we get:
n ≈ 5.25

Step 7: Interpret the result
Since n represents the number of terms in the series, it must be a positive whole number. Therefore, we can conclude that we need 6 terms of the series to reach the value 14 + 7√2.

Conclusion:
To reach the value 14 + 7√2, we need 6 terms of the series: √2, 2√2, 4√2, 8√2, 16√2, and 32√2.

There are six terms required to obtain a sum of (14 + 7√ 2).