If sum of the series $1+4x+7x^{2}+10x^{3}+....\infty,$ where $|x|

Question

If sum of the series $1+4x+7x^{2}+10x^{3}+....\infty,$ where $|x|<1$, is $\frac{35}{16}$. then x equals :

Accepted Answer

To solve the problem, we need to find the value of $ x $ for which the sum of the series $ 1 + 4x + 7x^2 + 10x^3 + \ldots $ converges to $\frac{35}{16}$.

### Explanation:

The given series is an arithmetic-geometric progression (AGP). The general term of the series can be expressed as $ a_n = (3n - 2)x^{n-1} $, where $ n $ starts from 1.

To find the sum of this series, we can use the formula for the sum of an AGP:

1. **Identify the series components:**
   - The arithmetic sequence is $ 1, 4, 7, 10, \ldots $ with the first term $ a = 1 $ and common difference $ d = 3 $.
   - The geometric sequence is $ 1, x, x^2, x^3, \ldots $.

2. **Sum of the AGP:**
   The sum $ S $ of an AGP can be found using the formula:
   $$
   S = \frac{a}{1-x} + \frac{dx}{(1-x)^2}
   $$
   where $ a $ is the first term of the arithmetic sequence, $ d $ is the common difference, and $ x $ is the common ratio of the geometric sequence.

3. **Apply the formula:**
   - Here, $ a = 1 $ and $ d = 3 $.
   - Substitute these into the formula:
   $$
   S = \frac{1}{1-x} + \frac{3x}{(1-x)^2}
   $$

4. **Set the sum equal to the given value:**
   $$
   \frac{1}{1-x} + \frac{3x}{(1-x)^2} = \frac{35}{16}
   $$

5. **Solve for $ x $:**
   - Multiply through by $(1-x)^2$ to clear the denominators:
   $$
   (1-x) + 3x = \frac{35}{16}(1-x)^2
   $$
   - Simplify and solve the resulting quadratic equation for $ x $.

6. **Verification:**
   - Solving the equation, we find that $ x = \frac{1}{5} $ satisfies the equation.
   - Substitute $ x = \frac{1}{5} $ back into the sum formula to verify:
     $$
     S = \frac{1}{1-\frac{1}{5}} + \frac{3 \times \frac{1}{5}}{(1-\frac{1}{5})^2} = \frac{5}{4} + \frac{3 \times \frac{1}{5}}{\left(\frac{4}{5}\right)^2} = \frac{5}{4} + \frac{15}{16} = \frac{35}{16}
     $$
   - This confirms that $ x = \frac{1}{5} $ is indeed the correct solution.

### Why Other Options are Incorrect:

1. **Option 1: $ x = \frac{2}{5} $**
   - Substituting $ x = \frac{2}{5} $ into the sum formula does not yield $\frac{35}{16}$. The calculations will show a different sum, indicating this is incorrect.

2. **Option 3: $ x = \frac{3}{5} $**
   - Similarly, substituting $ x = \frac{3}{5} $ results in a sum that does not match $\frac{35}{16}$. The series diverges or converges to a different value.

3. **Option 4: None of these**
   - Since we have verified that $ x = \frac{1}{5} $ satisfies the equation, this option is incorrect.

Thus, the correct answer is indeed Option 2: $ x = \frac{1}{5} $.