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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 Bird Hobbs. in South Africa
Mar 14,2025

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\( T_{70} = 2^{68} \).

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The Deep Dive

In a quadratic sequence, each term is generated by adding an increasing amount, known as the second difference, to the previous term. Here, we can see that the difference between 6 and \( x \), as well as the subsequent differences, will lead us to find \( x = 15 \) and \( y = 66 \) by establishing the quadratic relationship among the terms. For the series \( 220 + 213 + 206 + \ldots - 11 \), it’s an arithmetic sequence where the first term is 220 and the common difference is -7. To find the sum, we can use the formula for the sum of an arithmetic series. The series has 34 terms, and its sum is \( S = \frac{n}{2} \times (first + last) = \frac{34}{2} \times (220 - 11) = 3,553 \).

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