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Sia 🤖 1 day, 4 hours ago
The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time.
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Sia 🤖 2 days, 7 hours ago
1, A1, A2, A3, ----- Am, 31 are in AP.
a = 1
an = 31
{tex}a _ { m + 2 } = 314{/tex}
an = a + (n - 1)d
31 = a + (m + 2 - 1)d
{tex}d = \frac { 30 } { m + 1 }{/tex}
{tex}\frac { A 7 } { A _ { n - 1 } } = \frac { 5 } { 9 }{/tex} (Given)
{tex}\frac { 1 + 7 \left( \frac { 30 } { m + 1 } \right) } { 1 + ( m - 1 ) \left( \frac { 30 } { m + 1 } \right) } = \frac { 5 } { 9 }{/tex}
m = 1
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Gaurav Seth 3 days, 7 hours ago
uppose,
A={1,2,3,4,5,6}
B={2,4,6}
In this condition, every element in B is contained in A so,
B(A
But however, B is the set of even numbers and A is the set of natural numbers and thus they are very different in terms of n.
We write even number as 2n whereas natural number just n.
So, B doesn't belong to A.
Rahul Raj 3 days, 12 hours ago
Posted by Birendar Kumar 3 days, 22 hours ago
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Gaurav Seth 3 days, 14 hours ago
Any of the ways we can arrange things, where the order is important.
Example: You want to visit the homes of three friends ("a"), ("b") and ("c"), but haven't decided in what order. What choices do you have?
Answer: {a,b,c} {a,c,b} {b,a,c} {b,c,a} {c,a,b} {c,b,a}
Posted by Isha Suryan 3 days, 11 hours ago
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Sia 🤖 3 days, 11 hours ago
Let P(n) = a + ar + ar2 + ..... + arn-1{tex} = \frac{{a({r^n} - 1)}}{{r - 1}}{/tex}.
For n = 1
{tex}P(1) = a{r^{1 - 1}} = \frac{{a({r^1} - 1)}}{{r - 1}} \Rightarrow a = a{/tex}
{tex}\therefore {/tex} P(1) is true
Let P(n) be true for n = k
{tex}\therefore P(k) = a + ar + a{r^2} + .... + a{r^{k - 1}}{/tex}{tex} = \frac{{a({r^k} - 1)}}{{r - 1}}{/tex} .... (i)
For n = k + 1
R.H.S. {tex} = \frac{{a({r^{k + 1}} - 1)}}{{r - 1}}{/tex}
L.H.S. {tex} = \frac{{a({r^k} - 1)}}{{r - 1}} + a{r^k}{/tex} [ Using (i)]
{tex} = \frac{{a{r^k}}}{{r - 1}} - \frac{a}{{r - 1}} + a{r^k}{/tex}
{tex} = a{r^k} \cdot \left( {\frac{1}{{r - 1}}+1} \right) - \frac{a}{{r - 1}} = \frac{{a{r^{k + 1}}}}{{r - 1}} - \frac{a}{{r - 1}}{/tex}{tex} = \frac{{a{r^{k + 1}} - a}}{{r - 1}}{/tex}
{tex} = \frac{{a({r^{k + 1}} - 1)}}{{r - 1}}{/tex}
{tex}\therefore {/tex} P(k + 1) is true
Thus P(k) is true {tex} \Rightarrow {/tex} P(k + 1) is true
Hence by principle of mathematical induction, P(n) is true for all {tex}n \in N{/tex}.
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