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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.

Ask by Lee Bradley. in South Africa
Mar 14,2025

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- \( x = 13 \) - \( y = 70 \) - The sum of the series is 3553. - The series in sigma-notation is \( \sum_{k=0}^{33} (220 - 7k) \). - The total distance the ball will bounce is 285 m. - The series converges for \( -4 < t < 2 \). When \( t = 15 \), the series does not converge. - The 70th term is \( 2^{68} \).

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Bonus Knowledge

The quadratic sequence you presented can be approached by recognizing that the second difference will remain constant. By determining the sequence's first differences and then the second difference, you can find the missing values \( x \) and \( y \). If you observe the relationship, you'll find \( x = 14 \) and \( y = 66 \). As for the series \( 220 + 213 + 206 + \ldots - 11 \), it’s an arithmetic series with a first term of 220, a common difference of -7, and a last term of -11. You can calculate the number of terms using the formula for the nth term of an arithmetic sequence. The sum can be computed using the formula \( S_n = \frac{n}{2} (a + l) \), where \( a \) is the first term, and \( l \) is the last term. With some calculations, you’ll find that the total sum is 1,323. As for sigma-notation, this series can be expressed as \( \sum_{n=0}^{31} (220 - 7n) \).

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