Question

If t8=3 , t12=52 find 1st terms and Common difference

of AP​

Answer 1

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:

$\boxed{\bold{t_n = t + (n-1)d}}$  

where  

$t_n$ = last term, t = first term, n = no. of terms and d = common difference

We have,

$t_8 = 3$

∴ $t + (8-1)d = 3$

$\implies t + 7d = 3$ . . . . Equation 1

and

$t_1_2 = 52$

∴ $t + (12-1)d = 52$

$\implies t + 11d = 52$ . . . . Equation 2

On subtracting equation 2 from equation 1, we get

t + 11d = 52

t + 7d = 3

-  -         -

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 4d = 49

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∴ d = $\frac{49}{4}$

 

On substituting the value of d in equation 1, we get

$t + (7 \times \frac{49}{4} ) = 3$

$\implies t + \frac{343}{4} = 3$

$\implies t = 3 - \frac{343}{4}$

$\implies t = \frac{12 - 343}{4}$

$\implies t = - \frac{331}{4}$  

Thus,  

$\boxed{\bold{The \:1st\:term\:of\:the \:A.P.\: is\:\rightarrow \underline{\frac{49}{4}} }}\\\boxed{\bold{The \:common\:difference\:\:of\:the \:A.P. is\:\rightarrow \underline{-\frac{331}{4}} }}$  

 

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Answer 2

Step-by-step explanation:

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