If 5 times the 5th term of an AP is equal to 8 times its 8th term, find the 1th term of the ap
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Answer:
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Answer:
If 5 times the 5th term of an AP is equal
to 8 times its 8th term, show that the 13th
term is zero.
5timesofthe5thterm=8thtimes
ofthe8thterm
Tofind:↦
⇝13thtermoftheA.P.
\begin{gathered}\underline{ \mathfrak{ \: \: Suppose, \: \: }} \\ \\ \mathtt{ \rightsquigarrow \: First \: \: term \: \: of \: \: the \: \: A.P. = a} \\ \mathtt{ \rightsquigarrow \: Difference = d}\end{gathered}
Suppose,
⇝FirsttermoftheA.P.=a
⇝Difference=d
\begin{gathered}\underline{ \bold{ \: A.T.Q., \: }} \\ \\ \: \: \: \: \: \: \mathtt{ 5 \times T_5 = 8 \times T_8 }\\ \\ \mathtt{\Longrightarrow \: 5[ a + (5-1)d]=8 [ a + (8-1)d]} \\ \\ \mathtt{\Longrightarrow 5(a + 4d)= 8 (a+7d) }\\ \\ \mathtt{ \Longrightarrow 5a + 20 \: d = 8a + 56 \: d} \\ \\ \mathtt{ \Longrightarrow 5a - 8a = 56 \: d - 20 \: d}\\ \\ \mathtt{\Longrightarrow - 3a = 36 \: d}\\ \\ \mathtt{\Longrightarrow a = \frac{36 \: d}{ - 3}} \\ \\ \: \: \: \mathtt{ \therefore \: \: \underline{ \: a = - 12 \: d \: }}\end{gathered}
A.T.Q.,
5×T
5
=8×T
8
⟹5[a+(5−1)d]=8[a+(8−1)d]
⟹5(a+4d)=8(a+7d)
⟹5a+20d=8a+56d
⟹5a−8a=56d−20d
⟹−3a=36d
⟹a=
−3
36d
∴
a=−12d
13thterm,T
13
=a+[(13−1)d]
=a+12d
=(−12d)+12d
[Asa=−12d]
=−12d+12d
=0
∴
13thterm=0