Question

The 14th term of an AP is twice its 8th term. If the 6th term is -8, then find the sum of its first 20 terms.


Grade 10 maths
Lesson: AP

Answer 1

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Answer 2

Solution :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

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

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,

According to statement,

↝ 14ᵗʰ term of an AP is twice the 8ᵗʰ term.

$\rm :\longmapsto\:a + (14 - 1)d = 2\bigg(a + (8 - 1)d\bigg)$

$\rm :\longmapsto\:a + 13d = 2\bigg(a + 7d\bigg)$

$\rm :\longmapsto\:a + 13d = 2a + 14d$

$\bf\implies \:a = - \: d - - - (1)$

According to statement again,

↝ 6ᵗʰ term of an AP is - 8

$\rm :\longmapsto\:a + (6 - 1)d = - 8$

$\rm :\longmapsto\:a + 5d = - 8$

$\rm :\longmapsto\: - \: d + 5d = - 8 \: \: \: \: \: \: \: \: \: \{a = - \: d \}$

$\rm :\longmapsto\:4d \: = \: - \: 8$

$\bf\implies \:d \: = \: - \: 2$

$\bf\implies \:a \: = \: 2$

Now,

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of first n terms of an arithmetic sequence is,

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

Wʜᴇʀᴇ,

• Sₙ is the sum of first n terms.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Tʜᴜs,

We have,

• First term, a = 2

• Common Difference, d = - 2

• Number of terms, n = 20

Hence,

↝ Sum of first 20 term is,

$\rm :\longmapsto\:S_{20} \: = \dfrac{20}{2} \bigg(2 \times 2 + (20 - 1) \times ( - 2) \bigg)$

$\rm :\longmapsto\:S_{20} \: = 10 \times \bigg(4 - 38\bigg)$

$\rm :\longmapsto\:S_{20} \: = 10 \times \bigg( - 34\bigg)$

$\bf :\implies\:S_{20} \: = - \: 340$

Additional Information :-

1. If a, b, c are in AP, then 2b = a + c.

2. Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is from the end is

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

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term from the end

• l is the last term of the sequence.

• n is the no. of term.

• d is the common difference.