Question

For an A.P. t17 = 54 and t9 = 30 then find the first term a and
common difference d.​

Answer 1

Given:

For an A.P. t17 = 54 and t9 = 30

To find:

The first term a and common difference d

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,

$t_1_7 = 54$

∴ $t + (17 - 1)d = 54$

$\implies t + 16d = 54$ . . . . Equation 1

and

$t_9 = 30$

∴ $t + (9 - 1)d = 30$

$\implies t + 8d = 30$ . . . . Equation 2

On subtracting equation 2 from equation 1, we get

$t + 16d = 54$

$t + 8d = 30$

-   -         -

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$8d = 24$

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∴ $d = \frac{24}{8} = 3$

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

$t + (16\times 3) = 54$

$\implies t = 6$

Thus,

$\boxed{The \:first \:term \:is\:\rightarrow\bold{ \underline{6}}}$.

$\boxed{The \:common \:difference \:is\:\rightarrow\bold{ \underline{3}}}$.

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