Geometric sequences

9000073004

Level: 
B
Consider a geometric sequence \((a_{n})_{n=1}^{\infty }\). Let \(q\) be the quotient and \(s_{n}\) be the sum of the first \(n\) terms. Given \(a_{1} = 1\), \(a_{3} = 4\) and \(a_{2} < 0\), find the sum of the first four terms of the sequence.
\(s_{4} = -5\)
\(s_{4} = 15\)
\(s_{4} = 14\)
\(s_{4} = 8\)

9000073005

Level: 
B
Consider a geometric sequence \((a_{n})_{n=1}^{\infty }\). Let \(q\) be the quotient and \(s_{n}\) be the sum of the first \(n\) terms. Given \(a_{2} = 1\) and \(a_{3} = 10\), find the sum of the first four terms of the sequence.
\(s_{4} = 111.1\)
\(s_{4} = 99.9\)
\(s_{4} = 111\)
\(s_{4} = 100\)

9000073006

Level: 
B
Consider a geometric sequence \((a_{n})_{n=1}^{\infty }\). Let \(q\) be the quotient and \(s_{n}\) be the sum of the first \(n\) terms. Given \(a_{1} = 1\) and \(a_{4} = -8\), find the sum of the first five terms of the sequence.
\(s_{5} = 11\)
\(s_{5} = 31\)
\(s_{5} = 16\)
\(s_{5} = -16\)

9000073007

Level: 
B
Consider a geometric sequence \((a_{n})_{n=1}^{\infty }\). Let \(q\) be the quotient and \(s_{n}\) be the sum of the first \(n\) terms. Given \(a_{1} = -1\: 000\) and \(a_{2} = 100\), find the sum of the first four terms of the sequence.
\(s_{4} = -909\)
\(s_{4} = -900\)
\(s_{4} = 911\)
\(s_{4} = -911\)

1003109202

Level: 
C
A blouse is on sale and its price was reduced twice, each time by ten percent. The difference between its original and its final price is \( 133\,\mathrm{CZK} \). What was the blouse original price before the discounts?
\( 700\,\mathrm{CZK} \)
\( 665\,\mathrm{CZK} \)
\( 1\,330\,\mathrm{CZK} \)
\( 1\,400\,\mathrm{CZK} \)
\( 750\,\mathrm{CZK} \)

1003109203

Level: 
C
The half-life of Francium is \( 21 \) minutes. How long will it take before its weight decreases from \( 512\,\mathrm{g} \) to \( 1\,\mathrm{g} \)?
\( 3 \) hours \( 9 \) minutes
\( 2 \) hours \( 48 \) minutes
\( 3 \) hours \( 30 \) minutes
\( 2 \) hours \( 27 \) minutes
\( 3 \) hours \( 51 \) minutes

1003109204

Level: 
C
Candles are being lit at a commemoration in the square. Gradually, every \( 15 \) seconds, two candles are lit from one already burning candle. It also took \( 15 \) seconds to lit the first candle. After \( 4 \) minutes, all candles were lit. How many candles were finally burning in the square?
\( 65\,535 \)
\( 32\,767 \)
\( 32\,768 \)
\( 131\,070 \)
\( 131\,071 \)

1003109206

Level: 
C
In \( 2016 \), the population of the Czech Republic increased by \( 25\,000 \) to \( 10\,578\,820 \) inhabitants. How many inhabitants will there live in the Czech Republic at the end of the year \( 2026 \), if the percentage increase is the same every year?
\( 10\,832\,100 \)
\( 10\,828\,820 \)
\( 10\,846\,286 \)
\( 10\,831\,495 \)
\( 10\,603\,879 \)

1003109207

Level: 
C
In \( 2015 \), \( 8\,688 \) choir members sang in the largest gospel choir in Manila. Imagine, the conductor wrote an e-mail on January \( 1 \)st to three choir members. Each of them forwarded the e-mail to other three choir members on the next day and so on... On which day would all members receive the e-mail?
\( 8 \) January
\( 15 \) January
\( 2 \) February
\( 8 \) February
\( 12 \) January