# For each of the following sequences: @ identify if the sequence is arithmetic, geometric or quadratic’. Justify your response. @ assuming the first it

For each of the following sequences: @ identify if the sequence is arithmetic, geometric or quadratic’. Justify your response. @ assuming the first item of each sequence is a1, give an expression for aj. (In other words, find a formula for the i-th term in the sequence). @ if the sequence is arithmetic or geometric, compute the sum of the first 10 terms in the sequence $i2,-12,72,-432,2592,...$
$ii9,18,31,48,69,94,...$
$iii14,11.5,9,6.5,4,1.5,...$
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Step 1 The series is $2,-12,72,-432,2592,...$ check the ratio $\frac{-12}{2}=-6$
$\frac{72}{-12}=-6$
$\frac{-432}{72}=-6$ this is geometrix series with common ratio r, where $r=-6$ the sequence is geometric (b) the general term ${a}_{i}={a}_{1}{r}^{\left(}i-1\right)$ where ${a}_{1}$ is the first term , r is the common ratio. substitute the values ${a}_{i}=2\left(-6{\right)}^{\left(}i-1\right)$ hence, the expression for Step 2 the sum of first 10 terms is given by the formula $\frac{a\left(1-{r}^{n}\right)}{1-r}$ substitute the values to get the sum of first 10 terms ${S}_{n}=\frac{a\left(1-{r}^{n}\right)}{1-r}$
$=\frac{2\left(1-\left(-6{\right)}^{n}\right)}{1-\left(-6\right)}$
$=2\left(1-\left(-6{\right)}^{n}\right)1+6$
$=17276050$ hence, the sum of first 10 terms is given by 17276050