3.1 Given the linear pattern: \( 7 ; 2 ;-3 ; \ldots \) 3.1.1 Determine the general term, \( T_{n} \), of the linear pattern. 3.1.2 Calculate the value of \( T_{20} \). 3.1.3 Which term in the pattern has a value of -138 ?

The pattern provided is a linear sequence where each term decreases by 5. Starting from 7, the common difference \( d \) is -5. The general term can be expressed as \( T_n = 7 + (n - 1)(-5) \), simplifying to \( T_n = 12 - 5n \). To find \( T_{20} \), substitute \( n = 20 \) into the general term: \( T_{20} = 12 - 5(20) = 12 - 100 = -88 \). To determine which term has a value of -138, set \( T_n = -138 \): \( 12 - 5n = -138 \). Rearranging gives \( -5n = -150 \) or \( n = 30 \). So, the 30th term is -138! In case you're curious about sequences like this, they often arise in real-world situations such as predicting expenses over time, where each subsequent expense amount decreases in a fixed pattern over a set number of periods. They can also help us understand how to model real-life linear relationships, enabling us to make informed forecasting decisions! An interesting twist in sequence study is exploring Fibonacci or geometric sequences, which often capture our imaginations in art and nature. If you find patterns fascinating, books like "The Book of Numbers" or "Mathematics: A Very Short Introduction" could spark your interest further, diving into number theory and the beauty of numerical patterns!