Question

Find the i th term of the geometrie sequence 10,-20,40,dots.

Answer 1

Final Answer:

The
`\(i\)`th term of the geometric sequence
`\(10, -20, 40, \ldots\)` can be expressed as
`\(a_i = 10 * (-2)^(i-1)\)`.

Step-by-step explanation:

In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor. In this case, the first term
`\(a_1\) is 10`, and to find the common ratio, we divide any term by its preceding term:
`\((-20)/(10) = -2\)`. So, the common ratio
`(\(r\))` is -2. The general formula for the \(i\)th term (\(a_i\)) in a geometric sequence is given by
`\(a_i = a_1 * r^((i-1))\)`.

Now, substituting the given values, we get
`\(a_i = 10 * (-2)^(i-1)\)`. This expression accurately represents the \(i\)th term of the given geometric sequence. For example, when
`\(i = 1\), \(a_1 = 10 \`times
`(-2)^(1-1) = 10 \`times
`(-2)^0 = 10\)`, which is the first term. When
`\(i = 2\), \(a_2 = 10 \`times
`(-2)^(2-1) = 10 * (-2)^1 = -20\)`, matching the second term in the sequence.

The exponent \((i-1)\) in the formula is crucial as it accounts for the position of the term in the sequence. Each subsequent term is obtained by multiplying the previous term by the common ratio, and the exponent ensures the correct power of the ratio is applied at each step. Therefore, the general formula
`\(a_i = 10 * (-2)^(i-1)\)` accurately describes the
`\(i\)`th term of the geometric sequence.

Answer 2

The i-th term of the geometric sequence 10, -20, 40, ... can be found using the formula
`\(a_i = a_1 * r^((i-1))\)`, where
`\(a_1\)` is the first term, (r) is the common ratio, and
`\(i\)` is the term number.

Step-by-step explanation:

In the given sequence, the first term
`\(a_1\)` is 10, and the common ratio (r) can be found by dividing any term by its preceding term. Let's take the second and first terms:
`\(r = (-20)/(10) = -2\)`.

Now, we can use the formula
`\(a_i = a_1 * r^((i-1))\)` to find the i-th term. Substituting the values we have, the expression becomes
`\(a_i = 10 * (-2)^((i-1))\)`.

This formula gives us the i-th term for any position i in the sequence. For example, if you want the 4th term, substitute (i = 4) into the formula: \
`(a_4 = 10 * (-2)^((4-1)) = 10 * (-2)^3 = 10 * (-8) = -80\)`.

In conclusion, the i-th term of the geometric sequence is
`\(a_i = 10 * (-2)^((i-1))\)`, where
`\(i\)` represents the position of the term in the sequence.