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.

To find the values of \( x \) and \( y \) in the quadratic sequence, we need to identify the pattern. The first differences between consecutive terms are \( x - 6 \), \( 26 - x \), and \( 45 - 26 = 19 \). The second differences of a quadratic sequence should be constant, so the differences of the first differences can help us solve for \( x \) and \( y \). After setting up equations, we establish \( x = 14 \) and \( y = 66 \). Now for the series \( 220 + 213 + 206 + ... - 11 \), we note it is an arithmetic series where the first term \( a = 220 \), the common difference \( d = -7 \), and the last term \( l = -11 \). To find the number of terms \( n \), we solve \( -11 = 220 + (n-1)(-7) \), yielding \( n = 34 \). The sum formula \( S_n = \frac{n}{2} (a + l) \) gives \( S_{34} = \frac{34}{2} (220 - 11) = 3571 \). For sigma notation, it can be expressed as \( \sum_{k=0}^{33} (220 - 7k) \). Regarding the bouncing ball, starting from 15 m, it bounces and loses \( 10\% \) height each time, resulting in it rebounding to \( 90\% \) of its previous height. The total distance \( D \) includes the drop and all bounces, which follows a geometric series formula. After calculating the total rises and drops, it can be shown that \( D < 290 \) m. For convergence in the series involving \( t \), \( t \) must be less than the critical point defined by the series structure, leading us to converge for \( l < 3 \). When substituting \( t = 15 \), we calculate the sum to infinity using the formula for geometric series, yielding the limit of contribution to infinity given the constraints defined. Lastly, to find the \( 70^{th} \) term from the given \( S_{\infty} = 2^{x-1} + 3 \), recognition of relationships between sums and sequences will simplify the calculations to \( T_n = S_n - S_{n-1} \). After resolving the equations, this gives a final structured response of \( 2 \cdot 3^{69} \).