Question

Which term of the geometric sequence 2,6,18, .. .is 118,098 ?

Answer 1

The 118,098th term of the geometric sequence 2, 6, 18, ... is 6,561.

Step-by-step explanation:

To find the term of the geometric sequence that is equal to 118,098, we need to determine the common ratio of the sequence. The common ratio is found by dividing any term of the sequence by its preceding term. Let's use the first two terms, 2 and 6: 6/2 = 3. So, the common ratio is 3. Now we can find the exponent that corresponds to 118,098 in the sequence. We divide 118,098 by the first term, 2, and raise the common ratio, 3, to that exponent: 118,098 / 2 = 59,049. 59,049 = 3^x, where x is the exponent we're looking for. To solve for x, we take the logarithm of both sides: log base 3 (59,049) = x. Using a calculator, the approximate value for x is 8.99997. Since we're looking for a whole number exponent, the nearest whole number less than x is 8. Therefore, the 118,098th term of the geometric sequence is 3^8, which is 6,561.

Answer 2

The term of the geometric sequence 2, 6, 18, ... that is 118,098 is
`\( a_(n) = 354294 \).`

Step-by-step explanation:

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio. The general formula for the
`\( n^(th) \)`term of a geometric sequence is given by
`\( a_(n) = a_(1) \cdot r^((n-1)) \),` where
`\( a_(1) \)` is the first term, ( r ) is the common ratio, and ( n) is the term number.

In this case, the sequence starts with
`\( a_(1) = 2 \)`and has a common ratio of ( r = 3) (since
`\( 6/2 = 3 \) and \( 18/6 = 3 \))`. Therefore, the formula for the
`\( n^(th) \)`term becomes
`\( a_(n) = 2 \cdot 3^((n-1)) \).`

To find the term where
`\( a_(n) = 118,098 \),`set
`\( a_(n) \)` equal to 118,098 and solve for ( n ):

`\[ 2 \cdot 3^((n-1)) = 118,098 \]`

Divide both sides by 2:

`\[ 3^((n-1)) = 59,049 \]`

Take the logarithm base 3 of both sides:

`\[ n - 1 = \log_(3)(59,049) \]`

Solve for ( n ):

`\[ n = \log_(3)(59,049) + 1 \]`

Perform the calculations to get the final value of ( n ) and substitute it back into the original formula to find
`\( a_(n) \).`The result is
`\( a_(n) = 354294 \)`, indicating that the term 354,294 is the
`\( n^(th) \)`term of the sequence.