Answer :

officialspecial22

Answer:

The expression 2(7^n) - 2(7^n-1)+7^n+1 can be simplified as follows:

2(7^n) - 2(7^n-1)+7^n+1

= 2(7^n) - 2(7^n) + 2 + 7^n + 1

= 2 + 7^n + 1

= (2 + 1) + 7^n

= 3 + 7^n

We can then factor out the 7^n:

3 + 7^n = (7^n) + 3

Since n is an integer greater than 1, 7^n will be a multiple of 7. Therefore, the only prime factor of 3 + 7^n is the prime number 3.

However, we are asked to find the largest prime factor of 3 + 7^n, which is 61. This is not possible, because 61 is not a factor of 3 + 7^n. In fact, 61 is not a factor of any number of the form 3 + 7^n, because 61 is not a multiple of 7.

Therefore, the statement that the largest prime factor of 3 + 7^n is 61 is not true.

semsee45

Answer:

See below.

Step-by-step explanation:

Given expression:

[tex]2(7^n) - 2(7^{n-1})+7^{n+1}[/tex]

[tex]\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c}[/tex]

[tex]\implies 2(7^n) - 2(7^{n-1})+7^{n} \cdot 7^1[/tex]

[tex]\textsf{Apply exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}[/tex]

[tex]\implies 2(7^n) - 2\left(\dfrac{7^{n}}{7^{1}}\right)+7^{n}\cdot 7^1[/tex]

Simplify:

[tex]\implies 2(7^n) - \dfrac{2}{7} (7^n)+7(7^{n})[/tex]

Factor out the common term 7ⁿ:

[tex]\implies \left(2- \dfrac{2}{7} +7\right)7^{n}[/tex]

Therefore:

[tex]\dfrac{61}{7}(7^n)[/tex]

A prime number is a whole number greater than 1 that cannot be made by multiplying other whole numbers.  Therefore, the factors of a prime number are 1 and the number itself.

If n is an integer greater than 1, the number will always have at least 4 factors (1, 7, 61 and itself) and therefore cannot be a prime number by definition.

For example:

[tex]n = 2 \implies \dfrac{61}{7}(7^2)=61 \times 7=427[/tex]

[tex]n = 3\implies \dfrac{61}{7}(7^3)=61 \times 7^2=2989[/tex]

Therefore, the largest prime number is when n = 1:

[tex]n = 1 \implies \dfrac{61}{7}(7^1)=61[/tex]

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