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Worked solution

Find a geometric term and its first threshold crossing

A GP has first term 22 and common ratio 88. Find its seventh term. Find also the least positive integer nn such that its nth term exceeds 1 000 0001\,000\,000.

CJC · Sequences and Series · Example 10

The seventh term

Answer

u7=524288u_7=524288.

Identify the progression and substitute

a=2,r=8,un=2(8)n−1,u7=2(8)7−1=524288a=2,\quad r=8,\quad u_n=2(8)^{n-1},\qquad u_7=2(8)^{7-1}=524288

The source gives a=2,r=8 and the general term.

How did this part go?

The least index above one million

Answer

n=8n=8.

Method 1: solve the inequality with logarithms

2(8)n−1>1000000,8n−1>500000,(n−1)ln⁡8>ln⁡500000,n−1>ln⁡500000ln⁡8≈6.31,n>1+ln⁡500000ln⁡8≈7.31.\begin{aligned}2(8)^{n-1}&>1000000,\\8^{n-1}&>500000,\\(n-1)\ln8&>\ln500000,\\n-1&>\frac{\ln500000}{\ln8}\approx6.31,\\n&>1+\frac{\ln500000}{\ln8}\approx7.31.\end{aligned}

Since ln 8 is positive, division preserves the inequality. Keep the exact logarithmic cutoff before rounding. The source displays 6.31 and 7.31 as rounded cutoffs; the retained exact inequality justifies choosing the next integer.

Method 2: inspect the calculator table

n2(8n−1)comparison with 10000007524288<84194304>933554432>\begin{array}{c|r|c}n&2(8^{n-1})&\text{comparison with }1000000\\\hline7&524288&<\\8&4194304&>\\9&33554432&>\end{array}

Enter Y1=2*8^(X−1). The source requires this table when using the calculator method. Highlight the Y1 value to read the actual integer; the Teacher warns against copying the rounded display 4.19E6.

Choose the first admissible integer

u7<1000000<u8⟹nmin⁡=8u_7<1000000<u_8\quad\Longrightarrow\quad n_{\min}=8

The terms increase because a=2 is positive and r=8>1. Term 7 is below the threshold and term 8 above it, so no earlier positive index works.

How did this part go?

Common mistakes

The requested index is an integer. Retain exact table values: 4.19E6 is a rounded display, not the exact eighth term.

Solution notes

The calculator screenshot includes X=0 as a display sample; the sequence in this question starts at n=1.