Certainly! Let's solve the equation [tex]\(6^n + 6^{n+1} = 42\)[/tex] step-by-step.
Step 1: Simplify [tex]\(6^{n+1}\)[/tex]
Notice that [tex]\(6^{n+1}\)[/tex] can be written as [tex]\(6^n \cdot 6\)[/tex]. So the equation becomes:
[tex]\[6^n + 6^n \cdot 6 = 42\][/tex]
Step 2: Factor out [tex]\(6^n\)[/tex]
We can factor [tex]\(6^n\)[/tex] from both terms on the left-hand side:
[tex]\[6^n (1 + 6) = 42\][/tex]
Step 3: Simplify the expression inside the parenthesis
[tex]\[6^n (7) = 42\][/tex]
Step 4: Isolate [tex]\(6^n\)[/tex]
Divide both sides of the equation by 7:
[tex]\[6^n = \frac{42}{7}\][/tex]
Step 5: Simplify the right-hand side
[tex]\[6^n = 6\][/tex]
Step 6: Solve for [tex]\(n\)[/tex]
To find [tex]\(n\)[/tex], we observe that the bases are the same (6), so their exponents must be equal:
[tex]\[n = 1\][/tex]
Thus, the solution to the equation [tex]\(6^n + 6^{n+1} = 42\)[/tex] is:
[tex]\[n = 1\][/tex]
