Algebra: Sequence and Series Test-4

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Algebra: Sequence and Series Test-4

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Question 1

If the 4^th term of an arithmetic progression is 14 and the 12^th term is 70, then the first, term is:

A	–10
B	–7
C	+7
D	+10

Question 1 Explanation:

Let the common difference be d.
14+8d=70,
=> d=7.
First term is 14-3d=14-2 x 7=-7.
The correct option is (b)

Question 2

By adding the same constant to each of 31, 7, –1 a geometric progression results. The common ratio is:

A	$ \displaystyle 13$
B	$ \displaystyle 2\frac{1}{3}$
C	$ \displaystyle -12$
D	$ \displaystyle None\,\,of\,\,these$

Question 2 Explanation:

$ \begin{array}{l}(x+31)(x-1)={{(x+7)}^{2}}\\=>{{x}^{2}}+14x+49={{x}^{2}}+30x-31\\=>16x=80\\=>x=5\end{array}$

Question 3

The Value of
$ \displaystyle \begin{array}{l}\frac{3}{{{1}^{2}}{{.2}^{2}}}+\frac{5}{{{2}^{2}}{{.3}^{2}}}+\frac{7}{{{3}^{2}}{{.4}^{2}}}+\frac{9}{{{4}^{2}}{{.5}^{2}}}+\frac{11}{{{5}^{2}}{{.6}^{2}}}\\+\frac{13}{{{6}^{2}}{{.7}^{2}}}+\frac{15}{{{7}^{2}}{{.8}^{2}}}+\frac{17}{{{8}^{2}}{{.9}^{2}}}+\frac{19}{{{9}^{2}}{{.10}^{2}}}\,\,is\end{array}$

A	$ \displaystyle \frac{1}{100}$
B	$ \displaystyle \frac{99}{100}$
C	$ \displaystyle \frac{101}{100}$
D	$ \displaystyle 1$

Question 3 Explanation:

$ \begin{array}{l}\frac{3}{{{1}^{2}}{{.2}^{2}}}+\frac{5}{{{2}^{2}}{{.3}^{2}}}+\frac{7}{{{3}^{2}}{{.4}^{2}}}+\frac{9}{{{4}^{2}}{{.5}^{2}}}\\+\frac{11}{{{5}^{2}}{{.6}^{2}}}+\frac{13}{{{6}^{2}}{{.7}^{2}}}+\frac{15}{{{7}^{2}}{{.8}^{2}}}+\frac{17}{{{8}^{2}}{{.9}^{2}}}+\frac{19}{{{9}^{2}}{{.10}^{2}}}\,\\=\frac{4-1}{{{1}^{2}}{{.2}^{2}}}+\frac{9-4}{{{2}^{2}}{{.3}^{2}}}+\frac{16-9}{{{3}^{2}}{{.4}^{2}}}+\frac{25-16}{{{4}^{2}}{{.5}^{2}}}\\+\frac{36-25}{{{5}^{2}}{{.6}^{2}}}+\frac{49-36}{{{6}^{2}}{{.7}^{2}}}+\frac{64-49}{{{7}^{2}}{{.8}^{2}}}\\+\frac{81-64}{{{8}^{2}}{{.9}^{2}}}+\frac{100-81}{{{9}^{2}}{{.10}^{2}}}\\=\frac{1}{{{1}^{2}}}-\,\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{2}}}-\,\frac{1}{{{3}^{2}}}+\frac{1}{{{3}^{2}}}-\,\frac{1}{{{4}^{2}}}\\+\frac{1}{{{4}^{2}}}-\,\frac{1}{{{5}^{2}}}+\frac{1}{{{5}^{2}}}-\,\frac{1}{{{6}^{2}}}\\+\frac{1}{{{6}^{2}}}-\,\frac{1}{{{7}^{2}}}+\frac{1}{{{7}^{2}}}-\,\frac{1}{{{8}^{2}}}\\+\frac{1}{{{8}^{2}}}-\,\frac{1}{{{9}^{2}}}+\frac{1}{{{9}^{2}}}-\,\frac{1}{{{10}^{2}}}\\=1-\frac{1}{100}\\=\frac{99}{100}\end{array}$

Question 4

The value of
$ \displaystyle 1-\frac{1}{20}+\frac{1}{{{20}^{2}}}-\frac{1}{{{20}^{3}}}+.....$ Correct to 5 places f decimal is:

A	1.05
B	0.9
C	2.9
D	0.5

Question 4 Explanation:

The series is a G.P. and the common ratio is (-1/20) .
Thus the infinite GP sum is (20/21) = 0.9
Thus the correct option is (b)

Question 5

For all integral values of n,
the largest number that exactly divides each number of the sequence
$ \displaystyle \begin{array}{l}\left( n-1 \right)\,n\,\left( n+1 \right)\,,\,\,\\n\,\left( n+1 \right)\,\left( n+2 \right),\,\,\\\left( n+1 \right)\,\left( n+2 \right)\,\left( n+3 \right)\,.......is\end{array}$