Question

Consider an Arithmetic sequence 171, 167, 163.....
Is 0 a term of this sequence
How many positive terms are there in this sequence

Answer 1

Answer

0 is not a term of the given sequence.

There are 43 terms in the sequence.

$\bf\large\underline{Given}$

• The AP is 171 , 167 , 163 ,......

$\bf\large\underline{Task}$

• To find if 0 is the term if the term given AP
• To find the number of positive terms of the given AP

$\bf\large\underline{Solution}$

Given, AP is 171 , 167 , 163 , ...,..

Here ,

first term , a = 171

common difference ,

» d = 167 - 171

» d =-4

Let us consider 0 be the nth term of the given AP ,

$\sf \implies a + (n - 1)d = 0 \\\\ \sf\implies 171 + (n - 1)(-4 )= 0 \\\\ \sf\implies (n - 1)(-4) = -171 \\\\ \sf\implies n - 1 = \dfrac{-171}{-4} \\\\ \sf\implies n = \dfrac{171}{4} + 1 \\\\ \sf\implies n = \dfrac{171+4}{4} \\\\ \sf\implies n = \dfrac{175}{4}$

since the value of n is in fraction so 0 is not a term of the given AP

Though 0 is not a term of AP , it gives information about the positive and negetive terms , since it lies on the middle.

So , 175/4 = 43.75

But 176/4 = 44

Thus ,

$\sf 171+(44-1)(-4) \\\\ = 171 - 172 \\\\ = -1$

-1 is the first negative term of the AP , thus 44-1 = 43 is the number of positive terms of the given AP

Answer 2

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

Consider an Arithmetic sequence 171, 167, 163.....

Is 0 a term of this sequence. How many positive terms are there in this sequence.

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

★ Given that,

• AP series : 171,167,163.....

★ To find,

• 0 is a term of the series.
• No. of positive terms.

★ Let,

• a1 = 171
• a2 = 167

Common difference (d) = a2 - a1

$\bf\:\implies 167 - 171$

$\bf\:\implies - 4$

Hence,the common difference (d) is " -4 ".

★ Now,

Consider 0 as nth term of this AP series.

• a = 171
• d = - 4
• an = 0

By using nth term of AP formula...

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

• Substitute the values.

$\bf\:\implies 0 = 171 + (n - 1)(-4)$

$\bf\:\implies 0 = 171 - 4n + 4$

$\bf\:\implies 0 = 175 - 4n$

$\bf\:\implies 0 + 4n = 175$

$\bf\:\implies 4n = 175$

$\bf\:\implies n = \frac{175 }{4}$

$\bf\:\implies n = 43.75$

Therefore, the value of n is in decimals. So, 0 can't be a term of the AP.

Since, 0 can't be a term of this AP, but it gives the details of both positive & negative terms.

$\bf\:\implies \frac{176}{4}$

$\bf\:\implies 44$

Therefore, now we can take n as 44.

$\bf\:\implies 171 + (44 - 1)(-4)$

$\bf\:\implies 171 - 176 + 4$

$\bf\:\implies 175 - 176$

$\bf\:\implies - 1$

∴ " -1 " was the starting negative term of this AP.

So,

$\bf\:\implies 44 - 1$

$\bf\:\implies 43$

$\underline{\boxed{\bf{\purple{ \therefore There\:are\:43\:positive\:terms\:in\:the\:given\:AP.}}}}\:\orange{\bigstar}$

★ More Information :

How to find the Common difference (d)?

Let us take the AP Series which was given in the question.

AP : 171,167,163....

Let,

$\tt\:\implies a_{1} = 171$

$\tt\:\implies a_{2} = 167$

$\tt\:\implies a_{3} = 163$

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

$\bf\:\implies 167 - 171 = 163 - 167$

$\bf\:\implies - 4 = - 4$

↪ From the above we can see that the difference between the successive terms is same (constant) which is " - 4 ".

↪ 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.

___________________________

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

___________________________