Find Value of:-
`4+3 xx 4 +3 xx 4^(2) +3xx 4^(3) +3 xx 4^(4) +3 xx 4^(5) = `?
(a)`5 xx 4^(5)`
(b)`9xx 4^(4)`
(c)`4^(6)`
(d)`10 xx 4^(4)`

Options

A. `5 xx 4^(5)`
B. `9xx 4^(4)`
C. `4^(6)`
D. `10 xx 4^(4)`

Answer

To solve the expression \(4 + 3 \times 4 + 3 \times 4^2 + 3 \times 4^3 + 3 \times 4^4 + 3 \times 4^5\), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ 4 + 3 \times 4 + 3 \times 4^2 + 3 \times 4^3 + 3 \times 4^4 + 3 \times 4^5 \]

### Step 2: Factor Out Common Terms
Notice that \(4\) can be factored out from all the terms after the first:
\[ = 4 + 3(4 + 4^2 + 4^3 + 4^4 + 4^5) \]

### Step 3: Simplify the Expression Inside the Parentheses
The expression inside the parentheses is a geometric series with the first term \(4\) and a common ratio of \(4\). The sum of a geometric series can be calculated using the formula:
\[ S_n = a \frac{(r^n - 1)}{r - 1} \]
where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

In our case:
- \(a = 4\)
- \(r = 4\)
- \(n = 5\) (since we have \(4, 4^2, 4^3, 4^4, 4^5\))

Thus, the sum becomes:
\[ S = 4 \frac{(4^5 - 1)}{4 - 1} = 4 \frac{(4^5 - 1)}{3} \]

### Step 4: Substitute Back into the Expression
Now, substituting back, we have:
\[ = 4 + 3 \left(4 \frac{(4^5 - 1)}{3}\right) \]
The \(3\) cancels out:
\[ = 4 + 4(4^5 - 1) \]
\[ = 4 + 4 \times 4^5 - 4 \]
\[ = 4 \times 4^5 \]

### Step 5: Simplify Further
This simplifies to:
\[ = 4^6 \]

### Conclusion
Thus, the value of the expression is:
\[ \boxed{4^6} \]