ESTION 2 Consider the following quadratic sequence: \( 6 ; x ; 26 ; 45 ; y ; \ldots \) Determine the values of \( x \) and \( y \). 2 Given the following series: \( 220+213+206+\ldots-11 \) 2.2.1 Calculate the sum of the series. 2.2.2 Write the series in sigma-notation. 2.3 A batt is dropped from a beight of 15 m . It bounces back and loses \( 10 \% \) of its prev height on each bounce. Show that the total distance the ball will bounce cannot exi 290 m . 2.4 Given: \( 25\left(\frac{1-t}{3}\right)+5\left(\frac{1-1}{3}\right)^{2}+\left(\frac{1-1}{3}\right)^{3}+ \). \( \qquad \) 2.4.1 For which value(s) of \( l \) will the series converge? 2.4.2 If \( t=15 \), calculate the sum to infinity of the series if it exists. 2.5 The sum of the first \( n \) terms of a sequence is \( S_{\infty}=2^{x-1}+3 \). Deternine the \( 70^{\text {th }} \) term. Leave your answer in the form \( a . b^{\circ} \) where \( a, b \) and \( p \) at integers.
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To tackle the quadratic sequence \( 6, x, 26, 45, y, \ldots \), you can first recognize that the second differences should be constant since it is a quadratic sequence. Calculate the first differences: \( x - 6 \), \( 26 - x \), \( 45 - 26 = 19 \), and \( y - 45 \). By setting up and solving these equations, you'll find \( x = 15 \) and \( y = 66 \) to complete the sequence. For the series \( 220 + 213 + 206 + \ldots - 11 \), this is an arithmetic series with the first term \( a = 220 \), common difference \( d = -7 \), and last term \( l = -11 \). You can find the number of terms \( n \) using the formula \( n = \frac{l - a}{d} + 1 \), which gives \( n = 34 \). The sum \( S_n = \frac{n}{2} (a + l) = 3,543 \). In sigma notation, this series can be expressed as \( \sum_{k=0}^{33} (220 - 7k) \). Enjoy learning about the patterns in sequences and series!
