Question

The sum of first seven terms of an ap is 182.If its 4th term and 17th terms are in the ratio1;5 find the ap

Answer 1

Given: The sum of the first seven terms of an AP is 182 and that the 4th and 17th terms are in the ratio 1:5.

To find: The AP.

Answer:

It's given that the 4th and 17th terms are in the ratio 1:5. This means that:

$\tt \dfrac{a\ +\ 3d}{a\ +\ 16d}\ =\ \dfrac{1}{5}$

Cross - multiplying,

$\tt 5a\ +\ 15d\ =\ a\ +\ 16d\\\\\\5a\ -\ a\ =\ 16d\ -\ 15d\\\\\\4a\ =\ d$

Therefore, d = 4a.

Now, again, it's given that the sum of the first seven terms is 182. This means that:

$\tt S_7\ =\ \dfrac{n}{2}\ \times\ [2a\ +\ (n\ -\ 1)d]\\\\\\182\ =\ \dfrac{7}{2}\ \times\ [2a\ +\ (7\ -\ 1)(4a)]\\\\\\364\ =\ 7\ \times\ [2a\ +\ 6(4a)]\\\\\\52\ =\ 2a\ +\ 24a\\\\\\52\ =\ 26a\\\\\\\dfrac{52}{26}\ =\ a\\\\\\2\ =\ a$

Therefore, the first term is 2.

Now earlier, we obtained d = 4a. Using the value of a that we got,

d = 4 * 2

d = 8

Therefore, the AP is 2, 10, 18, 26, ... .

Answer 2

$\huge\underline\mathrm{GivEn:-}$

↪The sum of first seven terms of an ap is 182

↪The 4th term and 17th terms are in the ratio1;5

$\huge\underline\mathrm{To\:Find:-}$

Find the A.P.

$\huge\underline\mathrm{SoLution:-}$

It is given that the 4th and 17th terms of an A.P. are in the ratio 1:5

According to question :-

$</strong><strong>\</strong><strong>h</strong><strong>u</strong><strong>g</strong><strong>e</strong><strong>\frac{a + 3d}{a + 16d} = \frac{1}{5}$

Use cross product

$5a + 15d = a + 16d$

$5a - a = 16d - 15d$

$4a = d$

Thus, d = 4a -1)

Now, the sum of first 7 terms is 182

We know that, the formula for finding the sum n terms of an AP is :-

S7 = $\large{\sf{\frac{n}{2}}}$ [2a + (n-1) $\times$ d]

According to question :-

$\implies$182 = $\large{\sf{\frac{7}{2}}}$ [2a +(7-1)4a]___(d = 4a find above)

$\implies$182 = $\large{\sf{\frac{7}{2}}}$[2a + 6$\times$ 4a]

$\implies$182 = $\large{\sf{\frac{7}{2}}}$ (26a)

$\implies$182 = 7 $\times$ 13a

$\implies$a = $\huge{\sf{\frac{182}{13 \:x \:7}}}$

$\large\bf{a = 2}$

Now, to find d, put the value of a in equation 1)

$\implies$d = 4$\times$2

$\large\bf{d = 8}$

Therefore, the AP is 2,10,18,26,.....

$\huge\underline\mathrm{ThAnKs...}$