Question

In an AP, 6th term is half of the 4th term
and the 3rd term is 15. How many terms
are needed to give a sum of 66?​

Answer 1

Answer:

P gives the sum 66 . Either x = 4 or x = 11 . Therefore, Either 4 or 11 terms are required to get a sum 66 .

Explanation:

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Answer 2

$\rm \hookrightarrow\pink{Given}\begin{cases} \rm\mapsto \orange{{6}^{th} \: is \: half \: of \: {4}^{th} \: term} \\ \rm \mapsto \purple{{3}^{rd} \: term \: is \: 15} \end{cases}$

$\rm \hookrightarrow\red{Find}\begin{cases} \rm\mapsto \blue{how \: many \: terms \: needed \: to \: give \: sum \: of \: 66} \end{cases}$

Solution:

We have,

In an A.P.

$\rm Sixth \: term = \dfrac{fourth \: term}{2}$

Let, f1st term = a

Common difference = d

Then,

$\rm T_{6} = \dfrac{T_{4}}{2}$

$\rm a + 5d= \dfrac{a + 3d}{2} \: \: \: \: \: [ \therefore \: a_{n} = a + (n - 1)d]$

$\rm \implies 2(a + 5d)= a + 3d$

$\rm \implies 2a + 10d= a + 3d$

$\rm \implies 2a - a= 3d - 10d$

$\rm \implies a= - 7d$

Now, According to given question,

Third term = 15

$\rm a + 2d = 15$

$\rm \implies ( - 7d)+ 2d = 15$

$\rm \implies - 7d+ 2d = 15$

$\rm \implies - 5d= 15$

$\rm \implies d= \dfrac{ - 15}{5} = - 3$

$\rm \implies d= - 3$

Now use this value of d

$\rm a + 2d = 15$

$\rm \implies a+ 2( - 3) = 15$

$\rm \implies a - 6= 15$

$\rm \implies a = 15 + 6 = 21$

So,

a = 21

Now,

Sum of n terms = 66

$\rm S_n = \dfrac{n}{2} (2a + (n - 1)d)$

$\rm \implies 66= \dfrac{n}{2} (2a + (n - 1)d)$

$\rm \implies 66= \dfrac{n}{2} [2 \times 21 + (n - 1)( - 3)]$

$\rm \implies 66= \dfrac{n}{2} [42 - 3n + 3]$

$\rm \implies 66= \dfrac{n}{2} [45 - 3n]$

$\rm \implies 66= \dfrac{3n}{2} [15 - n]$

$\rm \implies 66 \times \dfrac{2}{3} = n [15 - n]$

$\rm \implies \dfrac{132}{3} = n [15 - n]$

$\rm \implies 44 = n [15 - n]$

$\rm \implies 44 = 15n - {n}^{2}$

$\rm \implies 44 + {n}^{2} - 15n = 0$

$\rm \implies {n}^{2} - 15n + 44= 0$

Now, Apply middle split term

$\rm \implies {n}^{2} - 11n - 4n + 44= 0$

$\rm \implies n(n - 11) - 4(n - 11)= 0$

$\rm \implies (n - 11) (n - 4)= 0$

If, n - 11 = 0, n = 11

If, n - 4 = 0, n = 4

So, after 4th term the sum of all terms will be zero