Question

The 2nd term and 6th term of an A.P. are 9 and 25 respectively, find
(i) the first term and the common difference
(ii) the 24th term
(iii) `n’ such that tn = 101

Answer 1

Step-by-step explanation:

hope its clear and easy to understand

Answer 2

$\sf\small\underline\purple{Given:-}$

$\sf{\implies The\:2nd\: term=9}$

$\sf{\implies The\:6th\: term=25}$

$\sf\small\underline\purple{To\: Find:-}$

$\sf{\implies The\: first\: term=?}$

$\sf{\implies The\: common\: difference=?}$

$\sf{\implies The\:24th\: term=?}$

$\sf{\implies T\:_{(n)}=101\:n=?}$

$\sf\small\underline\purple{Solution:-}$

• To calculate the common difference , first term and the value of n , at first we have to assume the first term of AP be a and common difference be d. Then set up equation as per the given clue in the question. After that we have to solve the equation by solving we will get the value of a and d. then calculate 24th term , first term and the value of n:-]

$\sf\small\underline\purple{Formula\: used:-}$

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

$\tt{\implies T\:_{(2)}=a+(2-1)d}$

$\tt{\implies a+d=9------(i)}$

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

$\tt{\implies T\:_{(6)}=a+(6-1)d}$

$\tt{\implies a+5d=25------(ii)}$

• Here, solving equations (i) and (ii):-

$\tt{\implies a+d=9}$

$\tt{\implies a+5d=25}$

• By solving we get , here:-

$\tt{\implies -4d=-16\implies d=4}$

• Putting the value of d=4 in eq (i):-

$\tt{\implies a+d=9}$

$\tt{\implies a+4=9}$

$\tt{\implies a=9-4\implies a=5}$

• Now calculate 24th term:-

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

$\tt{\implies T\:_{(24)}=5+(24-1)4}$

$\tt{\implies T\:_{(24)}=5+23*4}$

$\tt{\implies T\:_{(24)}=5+92}$

$\tt{\implies T\:_{(24)}=97}$

• Now n=? when Tn=101:-

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

$\tt{\implies 101=5+(n-1)4}$

$\tt{\implies 101-5=(n-1)4}$

$\tt{\implies 96=(n-1)4}$

$\tt{\implies n-1=24}$

$\tt{\implies n=24+1}$

$\tt{\implies n=25}$

$\sf\large{Hence,}$

$\sf{\implies The\: first\: term(a)=5}$

$\sf{\implies The\: common\: difference(d)=4}$

$\sf{\implies The\:24th\: term=97}$

$\sf{\implies T\:_{(n)}=101\:n=25}$