find the first terms and common difference of an A.P. whose t8=3 and t12=52
Answers
Answer:
1 and 8
Step-by-step explanation:
3x3 =9-1=8
12x5=60-8=52
Answer:
Given:
t8 = 3 & t 12 = 52
To find:
The first term and common difference
Solution:
The formula of the nth term of an A.P. is as follows:
\boxed{\bold{t_n = t + (n-1)d}}
t
n
=t+(n−1)d
where
t_nt
n
= last term, t = first term, n = no. of terms and d = common difference
We have,
t_8 = 3t
8
=3
∴ t + (8-1)d = 3t+(8−1)d=3
\implies t + 7d = 3⟹t+7d=3 . . . . Equation 1
and
t_1_2 = 52
∴ t + (12-1)d = 52t+(12−1)d=52
\implies t + 11d = 52⟹t+11d=52 . . . . Equation 2
On subtracting equation 2 from equation 1, we get
t + 11d = 52
t + 7d = 3
- - -
--------------------
4d = 49
-------------------
∴ d = \frac{49}{4}
4
49
On substituting the value of d in equation 1, we get
t + (7 \times \frac{49}{4} ) = 3t+(7×
4
49
)=3
\implies t + \frac{343}{4} = 3⟹t+
4
343
=3
\implies t = 3 - \frac{343}{4}⟹t=3−
4
343
\implies t = \frac{12 - 343}{4}⟹t=
4
12−343
\implies t = - \frac{331}{4}⟹t=−
4
331
Thus,
\begin{gathered}\boxed{\bold{The \:1st\:term\:of\:the \:A.P.\: is\:\rightarrow \underline{\frac{49}{4}} }}\\\boxed{\bold{The \:common\:difference\:\:of\:the \:A.P. is\:\rightarrow \underline{-\frac{331}{4}} }}\end{gathered}
The1sttermoftheA.P.is→
4
49
ThecommondifferenceoftheA.P.is→
−
4
331
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