Find the first financial threshold period

Model

\(\displaystyle \begin{gathered}V_0=1500\\V_{n+1}=1.05V_n\end{gathered}\)

An ideal investment starts at $1,500 and grows by 5% each year, with no other cash flows. After how many completed years does it first exceed $1,650?

Use this ideal $1,000 model at 5% per year. Move through whole years and find the first value exceeding $1,250. The table/readout displays approximate decimals; it does not round the underlying model to cents after every update

Which completed year first exceeds $1,250?

A threshold asks for the first index

A value that works is not enough: it must be the earliest allowed index. Build a table by repeated multiplication or evaluate the explicit rule at candidate indices. Keep non-negative integer indices

1. 1Translate the exact comparison Exceeds means strictly greater; falls below means strictly less. Equality does not satisfy either \(\displaystyle u_n>T\quad\text{or}\quad u_n<T\)
2. 2Start at index zero The initial value has index 0. The index counts multiplications, not the number of listed values \(\displaystyle u_n=aR^n\)
3. 3Check both neighbouring terms For these positive monotone sequences, show that the previous term fails and the proposed term succeeds. Check index 0 separately if it already satisfies the condition

Transfer the index to a financial model

The same first-crossing reasoning applies when a geometric term represents money. The answer to the first question is a completed period count, not the amount remaining. The rate and index must refer to the same period

1. 1Name the period The initial amount is at index 0. The index counts completed applications of the stated annual, quarterly or monthly update, not dollars or automatically years \(\displaystyle \begin{gathered}V_0=P\\V_n=PR^n\end{gathered}\)
2. 2Keep the model assumptions Use the supplied fixed-rate model without extra cash flows, fees, asset additions or disposal. A real investment return or resale value need not follow it indefinitely

For u(n)=7×2ⁿ, how do “first reaches 56 or more” and “first exceeds 56” differ?

Why the neighbouring check works here

Positive geometric growth or decay moves in one direction. If the preceding term fails the threshold and the next term succeeds, all earlier terms also fail. State this monotone setting rather than applying the shortcut to every sequence

1. 1State the scope of the argument Positive starts with R>1 increase, and positive starts with 0<R<1 decrease. Their monotonicity justifies the neighbouring-term check; that shortcut alone does not establish the first crossing for an oscillating sequence

In this increasing investment model, a student finds that year 5 exceeds the target. What additional comparison establishes that year 5 is the first crossing?

A quarter is not a year

A model growing by 2% per quarter already supplies its period rate. Use multiplier 1.02 for each quarter. If the first crossing is at quarter 4, that is one year, not four years. Do not divide 2% by 4 again

1. 1Convert a found count, not the rate again The automatic questions give the rate per stated period. Do not divide it again. Convert the period count to years only when asked \(\displaystyle \begin{gathered}\text{years}=\frac{\text{quarters}}4\\\text{years}=\frac{\text{months}}{12}\end{gathered}\)

Model

\(\displaystyle V_n=5000(1.02)^n\)

A fixed growth model starts at $5,000 and grows by 2% per quarter. Find the first completed quarter at which its value exceeds $5,400

Model

\(\displaystyle 4\text{ quarters per year}\)

The preceding model first exceeds the target after 4 quarters. How many years is that?

A different model first crosses after 18 months. What elapsed time does this represent?

Model

\(\displaystyle V_n=20000(0.8)^n\)

A $20,000 asset loses 20% of its current value each year in an ideal reducing-balance model. After how many completed years does its value first fall below $11,000?

A cents display can hide a crossing

This fixed boundary example uses an ideal $100 asset with multiplier 0.898. At year 3 its exact value is below $72.42, even though its nearest-cent report is $72.42. Comparing only the rounded display would miss that first crossing

1. 1Compare the ideal values Keep full model precision for the decision. Round a requested amount once at the end; an amount displayed to cents can hide which side of a threshold the ideal value lies on

An ideal model gives $72.4150792 at year 3. Is year 3 below the $72.42 threshold?

A sequence starts at u(0)=18 and doubles. What is the first non-negative index at which it exceeds 15?

A constant geometric sequence has u(0)=9 and ratio 1. When does it first exceed 10?

Write a financial threshold argument

Ungraded written task. Choose a positive starting amount and a fixed growth multiplier above 1 with a target above the start, or a reducing-balance multiplier between 0 and 1 with a positive target below the start. Define the period and initial index, state assumptions, and find the first whole-period crossing. Show the unrounded values immediately before and at it and explain why earlier periods fail. Report any requested amount with stated rounding

One possible written model

Model an $8,000 investment at 5% per year with no deposits, withdrawals or fees, and ask when it first exceeds $10,000. The model increases. Year 4 fails and year 5 succeeds, so all earlier years also fail. The first crossing is after five completed years. Its amount reported to cents is $10,210.25; that rounded report was not used to choose the year

1. 1Keep the model assumptions Use the supplied fixed-rate model without extra cash flows, fees, asset additions or disposal. A real investment return or resale value need not follow it indefinitely

Check your result

State the period and units. Check the initial amount, then compare the unrounded values immediately before and at the crossing. Equality does not satisfy a strict threshold. Explain why earlier periods fail and check whether the model can cross the target at all. Convert the period count to years only when asked