Question

if an ap 19th term is 52 and 38 term is 128 find the sum of the first 56 terms .

Answer 1

Here is the answer

Solution :

Let the first term of the A.P be 'a'
Let the common difference be 'd'

$\bf {t_{19} = 52 \: and \: t_{38} = 128}$

By using the formula,
$\bf {t_{n} = a + (n - 1) d}$

$\bf {t_{19} = a + (19 - 1)d} \\ \bf{t_{19} = a + 18d} \\ \bf{ 52 = a + 18d... \because t_{19 } = 52}$

Let this be equation (1)

Now,
Similarly,

$\bf{t_{38} = a + (38 - 1)d}$

$\bf{ t_{38} = a + 37d} \\ \bf {128 = a + 37d.. \because t_{38 } = 128 }$

Let this be equation (2)

Now ,
We got 2 equations,

Add equation (1) and (2),

a + 18 d = 52
+ a + 37 d = 128
_____________
2a + 55d = 180... eq. (3)

Now,
$\bf{ s_{n} = \frac{n}{2} (2a + (n - 1)d)}$

$\bf{s_{56} = \frac{56}{2} (2a + (56 - 1)d}$

$\bf{s_{56} = 28(2a + 55d)}$

$\bf{from \: 3...} \\ \bf {28 \times 180 = 5040}$

Therefore,
The sum of the first 56 terms is 5040.

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