Question

if 10 times he 10th term of an ap is equal to 15 times the 15th term show that its 25th term is zero​

Answer 1

$\large\underline{\sf{Solution-}}$

Given :-

↝ In an AP series,

• 10 times the 10ᵗʰ term = 15 times the 15ᵗʰ term.

To prove :-

• 25ᵗʰ term of AP is 0.

Calculations :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Tʜᴜs,

↝ 10ᵗʰ term and 15ᵗʰ is,

$\rm :\longmapsto\:a_{10} = a + (10 - 1)d = a + 9d$

and

$\rm :\longmapsto\:a_{15} = a + (15 - 1)d = a + 14d$

According to statement,

10 times the 10ᵗʰ term = 15 times the 15ᵗʰ term

$\rm :\longmapsto\:10a_{10} = 15a_{15}$

$\rm :\longmapsto\:10 \times (a + 9d) = 15 \times (a + 14d)$

$\rm :\longmapsto\:10a + 90d = 15a + 210d$

$\rm :\longmapsto\:10a + 90d - 15a - 210d = 0$

$\rm :\longmapsto\: - 5a - 120d = 0$

$\rm :\longmapsto\: - 5(a + 24d) = 0$

$\rm :\longmapsto\: a + 24d = 0$

$\rm :\longmapsto\: a + (25 - 1)d = 0$

$\bf\implies \:a_{25} = 0$

Hence, Proved

Additional Information :-

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms of AP.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.