Predicting What Comes Next: Exploring Sequences and Progressions: questions with answers

The first ten terms are $1, -1, 1, -1, 1, -1, 1, -1, 1, -1$. The terms at positions 1, 3, 5, 7 and 9 equal 1, so the answer is 5. Repeated values count as separate terms because they occupy different positions.

A sequence is an ordered list, and $t_1$ is the term in the first position. Reordering changes the first term from 2 to 8, so the reordered sequence has $t_1 = 8$. The subscript 1 gives the position, not the term's value.

Put $n = 300$ into the explicit rule: $u_{300} = 2 \times 300 - 1 = 599$. The rule gives the term from its position, without finding earlier terms.

Term positions are whole numbers. Since 12.5 is not a whole number, no term has position 12.5. The 13th term follows the 12th term directly.

The rule first applies at the third term, and working it there needs both the first term and the second. So two starting values are needed. That is why the chapter writes the Virahanka numbers with $V_1 = 1$ and $V_2 = 2$.

The rule adds 3 to each term to get the next, but it does not specify the first term. Starting with 1 gives $1, 4, 7, 10, \ldots$; starting with 10 gives $10, 13, 16, 19, \ldots$. Each different first term gives a different sequence. There are infinitely many choices for the first term, so infinitely many sequences obey the rule.

Subtract each term from the next: $4 - 1 = 3$, $9 - 4 = 5$, $16 - 9 = 7$ and $25 - 16 = 9$. The differences are 3, 5, 7 and 9. They are not all the same, so this sequence has no common difference and it is not an arithmetic progression. The terms keep growing, but they grow by a different amount each time.

An arithmetic progression needs the SAME difference between every pair of neighbouring terms. In 7, 11, 15, 19 the difference is 4 throughout. The squares have differences 3, 5, 7; the doubling list has 3, 6, 12; the triangular numbers have 2, 3, 4. All grow, and none of them grows evenly.

The balance has to grow by 3150, which is 7 deposits of 450, so 7 months. Counting the opening balance as the 1st term makes that balance the 8th term, and the position is not the number of months.

The rise needed is 300000, which is 12 lots of 25000, so it takes 12 years. That salary is the 13th term, as the starting salary is the 1st term. The term number is one more than the years passed.

The total is $1 + 2 + \dots + 22$. Two such triangles fit together into a rectangle of 22 by 23, which holds 506 marbles, so ONE triangle holds half of that, 253.

The seats number $21 + \dots + 60$, which is 1830 - 210 = 1620. There are 40 rows, but they are not rows 1 to 40: sliding the range down to start at 1 keeps the right NUMBER of terms and changes every one of them.

Only the ratio is raised to the power: stage 5 has $4 \times 3^{4} = 324$ squares. Tripling 4 first and then raising THAT to the power would multiply by 4 once at every stage.

The ratio comes from dividing, not subtracting: 24 divided by 6 is 4. Two more multiplications past 96 give $96 \times 4 \times 4 = 1536$. The first gap of 18 is not the rule; the gaps grow.

Multiplying a positive number by a number between 0 and 1 gives a smaller positive number. So each term is smaller than the term before it, and every term stays positive. The common ratio is still the same number at each step, so the sequence is a geometric progression throughout.