True False[1 Marks ]

Question 11 Mark

If the sum of n terms of a sequence is quadratic expression, then it always represents an $A.P.$

Answer

False.
We know that the sum of n terms of $A.P.$ is
$\text{S}_\text{n}=\frac{\text{n}}{2}(2\text{a}+(\text{n}-1)\text{d})=\frac{\text{n}}{2}(2\text{a}-\text{d}+\text{nd})$
$=\Big(\frac{2\text{a}-\text{d}}{2}\Big)\text{n}+\Big(\frac{\text{d}}{2}\Big)\text{n}^2$
Thus$, S_n$ is of type $An^2 + Bn.$
But general quadratic expression is of the form $An^2 + Bn + C.$
Thus, if the sum of n terms of a sequence is quadratic expression of type $An^2 + Bn + C,$ where $\text{C}\neq0,$ it does not represents sum of $A.P.$

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Question 21 Mark

Any term of an $A.P. ($except first$)$ is equal to half the sum of terms which are equidistant from it.

Answer

True.
Let a be the first term and d be the common difference of the $A.P.$
Consider any term ar of an $A.P.$
Now$, a_{r+m} = ar + (m - 1)d$
And $a_{r-m} = ar + (m - 1)(-d)$
$\therefore a_{r+m} + a_{r-m }= a_r + (m - 1)d + a_r + (m - 1)(-d)$
$\Rightarrow a_{r+m} + a_{r-m} =2a_r$
$\Rightarrow\text{a}_\text{r}=\frac{\text{a}_{\text{r}+\text{m}}+\text{a}_{\text{r}-\text{m}}}{2}$
Thus, any term of an $A.P. ($except first$)$ is equal to half the sum of terms which are equidistant from it.

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Question 31 Mark

The sum or difference of two G.P.s, is again a G.P.

Answer

False.
Solution:
Let two G.P.s are $\text{a},\text{ar}_1,\text{ar}_1^2,\text{ar}_1^3....;$ and $\text{b},\text{br}_2,\text{br}_2^2,\text{br}_2^3,....$
Now, sum of two G.P.s is $\text{a}+\text{b},(\text{ar}_1+\text{br}_2),\big(\text{ar}_1^2+\text{br}_2^2\big),....$
Clearly,
$\frac{\text{ar}_1+\text{br}_2}{\text{a + b}}\neq\frac{\text{ar}_1^2+\text{br}_2^2}{\text{ar}_1+\text{br}_2}$
Similarly, for difference of two G.P.s, we get
$\frac{\text{ar}_1+\text{br}_2}{\text{a + b}}\neq\frac{\text{ar}_1^2+\text{br}_2^2}{\text{ar}_1+\text{br}_2}$
So, the sum or difference of two G.P.s is not a G.P.

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Question 41 Mark

Two sequences cannot be in both $A.P.$ and $G.P.$ together.

Answer

True.
Let us consider $G.P, a, ar$ and $ar^2$
If it is in $A.P$ then $\text{ar}-\text{a}\neq\text{ar}^2-\text{ar}$
Hence$,$ the given statement is True.

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Question 51 Mark

Every progression is a sequence but the converse i.e., every sequence is also a progression need not necessarily be true.

Answer

True.
Solution:
Let us consider a sequence of prime number 2, 3, 5, 7, 11, ….
It is clear that this progression is a sequence but sequence is not a progression because it does not follow a specific pattern. Here, the given statement is True.

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