Question

3. Find, 100 is a term of the A.P. 25, 28, 31​

Answer 1

Step-by-step explanation:

Here an=100

an=a+(n-1)d

100=25+(n-1)3

75=(n-1)3

75/3=n-1

25=n-1

26=n

So 100 is a term of this Ap

And it is 26th term

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Answer 2

$\sf\huge\blue{\underline{\underline{ Question : }}}$

Find, 100 is a term of the A.P. 25, 28, 31...

$\sf\huge\blue{\underline{\underline{ Solution : }}}$

★ Given that,

• AP : 25,28,31....

★ To find,

• 100 is a term of the AP or not.

★ Let,

• a1 = 25
• a2 = 28.

Common difference (d) = a2 - a1

$\rm\:\implies d = 28 - 25$

$\rm\:\implies d = 3$

Hence, the common difference (d) = 3

★ Now,

• a = 28
• d = 3
• an = 100
• n = ?

By using nth term formula

$\tt\green{:\implies a_{n} = a + (n - 1)d }$

• Substitute the values.

$\sf\:\implies 100 = 28 + (n - 1)(3)$

$\sf\:\implies 100 = 28 + 3n - 3$

$\sf\:\implies 100 = 25 + 3n$

$\sf\:\implies 3n = 100 - 25$

$\sf\:\implies 3n = 75$

$\sf\:\implies n = \frac{75}{3}$

$\sf\:\implies n = 25$

$\underline{\boxed{\bf{\purple{ \therefore Hence,\:100\:is\:a\:term\:of\:this\:AP.}}}}\:\orange{\bigstar}$

★ Extra Information :

✒ How to find the common difference (d)?

Let us take the AP in the given question.

AP = 25,28,31.....

Now,

• $\bf\:a_{1} = 25$
• $\bf\:a_{2} = 28$
• $\bf\:a_{3} = 31$

Common difference (d) : a2 - a1 = a3 - a2

$\bf\:\implies 28 - 25 = 31 - 28$

$\bf\:\implies 3 = 3$

↪ From the above we can see that the difference between the successive terms is same (constant) which is 3.

↪ so we can say that the given sequence is in A.P.

↪ If the 1st term and the common difference 'd' is given then we can make an arithmetic sequence.

Hence,we can find the common difference (d) like this.

___________________________

$\boxed{\begin{minipage}{5 cm} AP Formulae : \\ \\ : \implies a_{n} = a + (n - 1)d \\ \\ :\implies S_{n} = \frac{n}{2} [ 2a + (n - 1)d ] \end{minipage}}$

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