Some Special Series - UP Board (UPMSP) STD 11 Science MATHS Questions

Q 1M.C.Q (1 Marks)1 Mark

If $\text{S}_\text{n}=\sum\limits^{\text{n}}_{\text{r}=1}\frac{1+2+2^2+\ ...\text{ Sum to r terms}}{2^{\text{r}}},$ then $S_n$ is equal to:

A
$2^{\text{n}}-\text{n}-1$
B
$1-\frac{1}{2^{\text{n}}}$
✓
$\text{n}-1-\frac{1}{2^{\text{n}}}$
D
${2^{\text{n}}}-1$

Answer: C.

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Q 2M.C.Q (1 Marks)1 Mark

The sum of the series $\frac{1}{\log_24}+\frac{1}{\log_44}+\frac{1}{\log_84}+\ ...\ +\frac{1}{\log_2{^\text{n}4}}\text{ is}:$

A
$\frac{\text{n}(\text{n}+1)}{2}$
B
$\frac{\text{n}(\text{n}+1)(2\text{n}+1)}{12}$
✓
$\frac{\text{n}(\text{n}+1)}{4}$
D
none of these.

Answer: C.

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Q 3M.C.Q (1 Marks)1 Mark

Sum of $n$ terms of the series $\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+\ ...\text{ is}$

A
$\frac{\text{n}(\text{n}+1)}{2}$
B
$2\text{n}(\text{n}+1)$
✓
$\frac{\text{n}(\text{n}+1)}{\sqrt{2}}$
D
$1$

Answer: C.

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Q 4M.C.Q (1 Marks)1 Mark

The value of $\sum\limits^{\text{n}}_{\text{r}=1}\Big\{\big(2\text{r}-1\big)\text{a}+\frac{1}{\text{b}^\text{r}}\Big\}$ is equal to:

A
$\text{an}^2+\frac{\text{b}^{\text{n}-1}-1}{\text{b}^{\text{n}-1}(\text{b}-1)}$
✓
$\text{an}^2+\frac{\text{b}^\text{n}-1}{\text{b}^\text{n}(\text{b}-1)}$
C
$\text{an}^3+\frac{\text{b}^{\text{n}-1}-1}{\text{b}^{\text{n}}(\text{b}-1)}$
D
none of these.

Answer: B.

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Q 5M.C.Q (1 Marks)1 Mark

The sum of $10$ terms of the series $\sqrt{2}+\sqrt{6}+\sqrt{18}+ ....$ is

✓
$121\big(\sqrt{6}+\sqrt{2}\big)$
B
$243\big(\sqrt{3}+1\big)$
C
$\frac{121}{\sqrt{3}-1}$
D
$242\big(\sqrt{3}-1\big)$

Answer: A.

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Q 61 Marks Question1 Mark

Write the sum of the series $1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + ..... + (2n - 1)^2 - (2n)^2$

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Q 71 Marks Question1 Mark

If the sum of first $n$ even natural numbers is equal to $k$ times the sum of first n odd natural numbers, then write the value of $k.$

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Q 81 Marks Question1 Mark

Let $S_n$ denote the sum of the cubes of first n natural numbers and $s_n$ denote the sum of first $n$ natural numbers. Then, write the value of $\sum\limits^\text{n}_{\text{r}=1}\frac{\text{S}_\text{r}}{\text{S}_\text{r}}$

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Q 91 Marks Question1 Mark

If $\sum\limits^{\text{n}}_{\text{r}=1}\text{r}=55,$ find $\sum\limits^{\text{n}}_{\text{r}=1}\text{r}^3$

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Q 101 Marks Question1 Mark

Write the sum of the series $2 + 4 + 6 + 8 + ... + 2n$.

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Q 115 Marks Questions5 Marks

Find the sum of the following series to n terms:
$1.2.4 + 2.3.7 +3.4.10 + ...$

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Q 125 Marks Questions5 Marks

Find the sum of the series whose $n^{th} $ term is:
$2n^3 + 3n^2 - 1$