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Sequences and Series

Multiple Choice Questions

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is:

46^th
59^th
52^nd
52^nd

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If the 2^nd, 5^th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

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If the sum of the first ten terms of the series,

$open parentheses 1 3 over 5 close parentheses squared space plus open parentheses 2 2 over 5 close parentheses squared space plus open parentheses 3 1 fifth close parentheses squared space plus space 4 squared space plus space open parentheses 4 4 over 5 close parentheses squared space plus.....$ is 16/5 m, the m is equal to

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The sum of first 9 terms of the series $1 cubed over 1 space plus fraction numerator 1 cubed plus 2 cubed over denominator 1 plus 3 end fraction space plus space fraction numerator 1 cubed plus 2 cubed plus 3 cubed over denominator 1 plus 2 plus 3 end fraction space....$ is

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Three positive numbers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. Then, the common ratio of the GP is

$square root of 2 plus square root of 3$
$3 plus square root of 2$
$2 minus square root of 3$
$2 minus square root of 3$

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The sum of first 20 terms of the sequence 0.7,0.77,0.777...... is

$7 over 81 space left parenthesis 179 minus 10 to the power of negative 20 end exponent right parenthesis$
$7 over 9 left parenthesis 99 minus 10 to the power of negative 20 end exponent right parenthesis$
$7 over 81 left parenthesis 179 plus 10 to the power of negative 20 end exponent right parenthesis$
$7 over 81 left parenthesis 179 plus 10 to the power of negative 20 end exponent right parenthesis$

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If x, y, z are in A.P. and tan⁻¹ x, tan⁻¹ y and tan⁻¹ z are also in A.P., then

x= y= z
2x =3y = 6z
6x = 3y= 2z
6x = 3y= 2z

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8.
Statement 1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + ...... + (361 + 380 +400) is 8000.
Statement 2: $sum from straight k space equals 1 to straight n equals space 1 of left square bracket straight k cubed minus left parenthesis straight k minus 1 cubed right parenthesis right square bracket space equals space straight n cubed$ , for any natural number n.

Statement 1 is false, statement 2 is true

Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

Statement 1 has 20 terms whose sum is 8000 And statement 2 is true and supporting statement 1.
k^th bracket is (k – 1)² + k(k – 1) + k² = 3k² – 3k + 1.

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If 100 times the 100^th term of an AP with non zero common difference equals the 50 times its 50^th term, then the 150^th term of this AP is

–150
150
times its 50th term
times its 50th term

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10.

Let x1, x2, ......, x_n be n observations, and let $top enclose straight x$ be their arithmetic mean and σ2 be their variance.
Statement 1: Variance of 2x1, 2x2, ......, 2xn is 4 σ².
Statement 2: Arithmetic mean of 2x1, 2x2, ......, 2xn is 4 $top enclose straight x$ .

Statement 1 is false, statement 2 is true
Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

213 Views