What Is The 11th Term Of The Geometric Sequence 2 6?

The 11th term of the geometric sequence starting with 2 and 6 is 118,098.

This result is found by first determining the common ratio of the sequence, and then using the explicit formula for the nn

𝑛

-th term of a geometric sequence.

Overview of geometric sequences

A geometric sequence, or geometric progression, is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

• **First term (a1a sub 1

  𝑎1

  )**: The first number in the sequence. In this case, a1=2a sub 1 equals 2

  𝑎1=2

  .

• **Common ratio (rr

  𝑟

  )**: The constant number that each term is multiplied by to get the next. It is found by dividing any term by its preceding term.

• Explicit formula: The formula to find any nn

  𝑛

  -th term of the sequence is an=a1⋅r(n−1)a sub n equals a sub 1 center dot r raised to the open paren n minus 1 close paren power

  𝑎𝑛=𝑎1⋅𝑟(𝑛−1)

  .

Step-by-step calculation

Step 1: Find the common ratio (rr

𝑟

)

To find the common ratio, divide the second term by the first term.

• Second term: 6

• First term: 2

• r=62=3r equals six-halves equals 3

  𝑟=62=3

Step 2: Write the explicit formula

With the first term and the common ratio, you can write the explicit formula for this specific geometric sequence:an=2⋅3(n−1)a sub n equals 2 center dot 3 raised to the open paren n minus 1 close paren power

𝑎𝑛=2⋅3(𝑛−1)

Step 3: Solve for the 11th term

To find the 11th term, substitute n=11n equals 11

𝑛=11

into the explicit formula:

1. Set up the equation: a11=2⋅3(11−1)a sub 11 equals 2 center dot 3 raised to the open paren 11 minus 1 close paren power

   𝑎11=2⋅3(11−1)

   .

2. Simplify the exponent: a11=2⋅310a sub 11 equals 2 center dot 3 to the tenth power

   𝑎11=2⋅310

   .

3. Calculate the power: 310=59,0493 to the tenth power equals 59 comma 049

   310=59,049

   .

4. Multiply by the first term: a11=2⋅59,049=118,098a sub 11 equals 2 center dot 59 comma 049 equals 118 comma 098

   𝑎11=2⋅59,049=118,098

   .

Discussion of results

The resulting value of 118,098 demonstrates the exponential growth characteristic of geometric sequences. Unlike arithmetic sequences, which have a constant difference between terms, geometric sequences have a constant ratio, leading to much faster growth. This concept has broad applications in mathematics and science, including calculating compound interest, modeling population growth, and understanding radioactive decay.

The derivation of the formula an=a1⋅r(n−1)a sub n equals a sub 1 center dot r raised to the open paren n minus 1 close paren power

𝑎𝑛=𝑎1⋅𝑟(𝑛−1)

is rooted in the definition of a geometric sequence. Each successive term is found by multiplying the previous one by rr

𝑟

:

• a2=a1⋅ra sub 2 equals a sub 1 center dot r

  𝑎2=𝑎1⋅𝑟

• a3=a2⋅r=(a1⋅r)⋅r=a1⋅r2a sub 3 equals a sub 2 center dot r equals open paren a sub 1 center dot r close paren center dot r equals a sub 1 center dot r squared

  𝑎3=𝑎2⋅𝑟=(𝑎1⋅𝑟)⋅𝑟=𝑎1⋅𝑟2

• a4=a3⋅r=(a1⋅r2)⋅r=a1⋅r3a sub 4 equals a sub 3 center dot r equals open paren a sub 1 center dot r squared close paren center dot r equals a sub 1 center dot r cubed

  𝑎4=𝑎3⋅𝑟=(𝑎1⋅𝑟2)⋅𝑟=𝑎1⋅𝑟3

  Following this pattern, the exponent on rr

  𝑟

  is always one less than the term number, which leads to the general formula for the nn

  𝑛

  -th term.