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1003097008

Level:

Find the first term of an arithmetic sequence $ \left\{a_n\right\}_{n=1}^{\infty} $, if the sum of the first eight terms is $12$ and the sum of the first twelve terms is $-6$: \[ \begin{aligned} s_8&=12 \\ s_{12}&=-6 \end{aligned} \]

$ 5 $

$ -5 $

$ 2 $

$ 1 $

$ -3 $

1003097007

Level:

Find the common difference of an arithmetic sequence $ \left\{a_n\right\}_{n=1}^{\infty} $, if the sum of the first seven terms is $42$ and $a_{10}=-4a_5$: \[ \begin{aligned} s_7&=42 \\ a_{10}&=-4a_5 \end{aligned} \]

$ -3 $

$ 3 $

$ 15 $

$ 2 $

$ -2 $

1003097006

Level:

Find the sum of the first ten terms of an arithmetic sequence $ \left\{a_n\right\}_{n=1}^{\infty} $, if: \[ \begin{aligned} a_1+a_5+a_{10}&=40 \\ a_{10}-a_5-a_1&=24 \end{aligned} \]

$ 140 $

$ 180 $

$ 160 $

$ 320 $

$ 200 $

1003097005

Level:

Find the square of the first term of an arithmetic sequence $ \left\{a_n\right\}_{n=1}^{\infty} $, if: \[ \begin{aligned} a_2^2+a_3^2&=100 \\ a_5+a_7&=0 \end{aligned} \]

$ 100 $

$ 196 $

$ 64 $

$ 169 $

$ 200 $

1003097004

Level:

Find the third term of an arithmetic sequence $ \left\{a_n\right\}_{n=1}^{\infty} $, if: \[ \begin{aligned} 2a_{10}-3a_3&=5 \\ 5a_5+4a_1&=83 \end{aligned} \]

$ 9 $

$ 7 $

$ 1 $

$ 3 $

$ 11 $

1003097003

Level:

Find the sum of the first five terms of an arithmetic sequence $ \left\{a_n\right\}_{n=1}^{\infty} $, if: \[ \begin{aligned} a_1\cdot a_4&=100 \\ a_2+a_6&=10 \end{aligned} \]

$ 50 $

$ 0 $

$ 20 $

$ -5 $

$ 90 $

1003097002

Level:

Find the common difference of an arithmetic sequence $ \left\{a_n\right\}_{n=1}^{\infty} $, if: \[ \begin{aligned} a_{11}+a_4&=-19 \\ a_6+a_7&=-13 \end{aligned} \]

$ -3 $

$ 3 $

$ 7 $

$ -1 $

$ 10 $

1003097001

Level:

Find the first term of an arithmetic sequence $ \left\{a_n\right\}_{n=1}^{\infty} $, if: \[\begin{aligned} a_2+a_9&=24 \\ a_8-a_6&=4 \end{aligned} \]

$ 3 $

$ 6 $

$ 2 $

$ 10 $

$ 1 $

1003112812

Level:

The sum $ s_3 $ of the first three terms of a geometric sequence is $ 21 $ and its common ratio is $ -2 $. Which of the following (in)equalities is true for the first term of the sequence?

$ a_1 > 0 $

$ a_1 < 0 $

$ a_1 = 0 $

$ a_1 < a_2 $

$ a_1 > s_3 $

1003112810

Level:

The sum of the first $ n $ terms of a geometric sequence is $ -5 $, the common ratio is $ -2 $ and the first term equals $ 1 $. Find $ n $.

$ 4 $

$ 3 $

$ 5 $

$ 6 $

$ 2 $

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