Question

How many terms of the AP 24 21 18 must be taken so that their sum is 78

Answer 1

Answer:

hope this is helpful.

Answer 2

Given :-

• a = 24

• d = 21 - 24 = -3

• $\rm\:S_n$= 78

To find :-

• Number of terms (n) = ?

Solution :-

We know that,

$\purple{\bigstar}\:\:{\underline{\boxed{\bf\red{S_n=\dfrac{n}{2}\:[2a+(n-1)d]}}}}$

$\rm\longrightarrow\:78=\dfrac{n}{2}\:[2\times\:24+(n-1)(-3)]$

$\rm\longrightarrow\:78=\dfrac{n}{2}\:[48+(n-1)(-3)]$

$\rm\longrightarrow\:78=\dfrac{n}{2}\:[51-3n]$

$\rm\longrightarrow\:78\times\:2=51n-3{n}^{2}$

$\rm\longrightarrow\:156=51n-3{n}^{2}$

$\rm\longrightarrow\:3{n}^{2}-51n+156=0$

$\rm\longrightarrow\:3({n}^{2}-17+52)=0$

$\rm\longrightarrow\:{n}^{2}-17n+52=0$

$\rm\longrightarrow\:{n}^{2}-13n-4n+52=0$

$\rm\longrightarrow\:n(n-13)-4(n-13)=0$

$\rm\longrightarrow\:(n-4)(n-13)=0$

Now,

$\rm\longrightarrow\:n-4=0$

$\rm\longrightarrow\:n=0+4$

$\rm\longrightarrow\:n=4$

Then,

$\rm\longrightarrow\:n-13=0$

$\rm\longrightarrow\:n=0+13$

$\rm\longrightarrow\:n=13$

Hence,the number of terms (n) will be either 4 or 13.