Find the sum of n terms of the sequence : 5 + 55 + 555 + ...........

Answer

To find the sum of the first n terms of the sequence: 5 + 55 + 555 + ..., we can follow these steps:

### Step 1: Express the Terms
The given sequence can be expressed as:
- First term (T1) = 5
- Second term (T2) = 55
- Third term (T3) = 555
- And so on...

We can see that each term can be represented as:
- T1 = 5
- T2 = 5 × 11
- T3 = 5 × 111
- T4 = 5 × 1111
- ...

### Step 2: Factor Out the Common Term
We can factor out 5 from each term:
\[ S_n = 5 + 55 + 555 + ... = 5(1 + 11 + 111 + ... ) \]

### Step 3: Rewrite the Inner Sum
Now, we need to find the sum of the series inside the parentheses:
\[ 1 + 11 + 111 + ... \]

This can be rewritten as:
\[ 1 + 11 + 111 + ... = 1 + (10 + 1) + (100 + 10 + 1) + ... \]
This can be expressed as:
\[ = 1 + 10 + 100 + ... + 10^{n-1} + n \]
Where n is the number of terms.

### Step 4: Use the Formula for the Sum of a Geometric Series
The series \( 1 + 10 + 100 + ... + 10^{n-1} \) is a geometric series where:
- First term (a) = 1
- Common ratio (r) = 10
- Number of terms (n)

The sum of the first n terms of a geometric series can be calculated using the formula:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]
Substituting the values:
\[ S_n = \frac{1(10^n - 1)}{10 - 1} = \frac{10^n - 1}{9} \]

### Step 5: Substitute Back
Now, substituting this back into our expression for S:
\[ S_n = 5 \left( \frac{10^n - 1}{9} \right) \]
\[ S_n = \frac{5(10^n - 1)}{9} \]

### Final Result
Thus, the sum of the first n terms of the sequence is:
\[ S_n = \frac{5(10^n - 1)}{9} \]

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