Comparing sequences: nth terms, graphs and intersection

Task goals

• Find the nth term rule of a linear sequence from its first terms, using the common difference (dn + c form), including decreasing sequences.
• Plot the terms of two sequences as points on the same axes (term number n horizontal, term value vertical) and draw or extend the lines through them.
• Read the intersection point of two plotted sequences from a graph, and state the term number and term value at which they are equal.
• Verify an intersection algebraically by setting the two nth-term rules equal and solving for n.
• Interpret the comparison in context, stating from which term onwards one sequence is larger, including when the algebraic solution for n is not a whole number (so no term is equal and the overtake happens between consecutive terms).
• Compare a simple quadratic sequence (n squared plus a constant) with a linear sequence and explain which is eventually larger and from which term.