(iii) She did not release him.
Answers
Answer:
It is given that:
\sf \bullet a_7=20∙a
7
=20
\sf \bullet a_{2n}=a_2a_n+1∙a
2n
=a
2
a
n
+1
\sf\bullet a_{2n+1}=a_2a_n-2∙a
2n+1
=a
2
a
n
−2
To find:
\sf\bullet a_{25}∙a
25
To find the 25th term, we have to find the value of common difference and first term.
We can see that \sf a_{2n}a
2n
and \sf a_{2n+1}a
2n+1
will be 2 consecutive terms of AP. So we can find common difference by subtracting these terms.
So:-
\sf :\Longrightarrow a_{2n+1}-a_{2n}=common\: difference:⟹a
2n+1
−a
2n
=commondifference
\sf :\Longrightarrow a_{2}a_n-2-(a_{2}a_n+1)=common\: difference:⟹a
2
a
n
−2−(a
2
a
n
+1)=commondifference
\sf :\Longrightarrow a_{2}a_n-2-a_{2}a_n-1=common\: difference:⟹a
2
a
n
−2−a
2
a
n
−1=commondifference
\sf :\Longrightarrow -2-1=common\: difference:⟹−2−1=commondifference
\sf :\Longrightarrow -3=common\: difference:⟹−3=commondifference
So the common difference of the AP is -3.
Now,
\sf :\Longrightarrow a_7=20:⟹a
7
=20
\sf :\Longrightarrow a+6d=20:⟹a+6d=20
Here d=common difference
By putting value of common difference:
\sf :\Longrightarrow a+6(-3)=20:⟹a+6(−3)=20
\sf :\Longrightarrow a-18=20:⟹a−18=20
\sf :\Longrightarrow a=20+18:⟹a=20+18
\sf :\Longrightarrow a=38:⟹a=38
Now find 25th term:
\sf :\Longrightarrow a_{25}=a+24d:⟹a
25
=a+24d
By putting values of a and d:
\sf :\Longrightarrow a_{25}=38+24(-3):⟹a
25
=38+24(−3)
\sf :\Longrightarrow a_{25}=38-72:⟹a
25
=38−72
:\implies \boxed{\sf{ a_{25}=-34}}:⟹
a
25
=−34
So the required \bf a_{25}a
25
is -34.