Question

if t⁴=10 and t⁸=30 d=??
ap ​

Answer 1

Answer:

d=5

Step-by-step explanation:

${t}^{4} = 10 \\ t + 3d = 10 \: \: \: \: ...(1) \\ and \: \: \: \: t + 7d = 30 \: \: \: ...(2) \\ subtract \: eq(2) \: \: from \: eq(1) \: \: - \\ \: t + 7d - t - 3d = 30 - 10 \\ 4d = 20 \\ d = 5 \: \: \: \: \: \: \: (answer)$

Answer 2

Given Question :-

In an AP series,

$\rm \: t_4 = 10 \: \: and \: \: t_8 = 30, \: \: find \: d$

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

Given that,

In an AP series,

$\rm \: t_4 = 10 \: \\ \\ \rm \: \: t_8 = 30 \:$

Let assume that

First term of an AP series is a

Common difference of an AP series is d

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

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

Wʜᴇʀᴇ,

tₙ is the nᵗʰ term.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.

Tʜᴜs,

$\rm \: t_4 = 10$

$\rm \: a + (4 - 1)d = 10$

$\rm\implies \:a + 3d = 10 - - - (1)$

and

$\rm \: t_8 = 30$

$\rm \: a + (8 - 1)d = 30$

$\rm\implies \:a + 7d = 30 - - - (2)$

On Subtracting equation (1) from equation (2), we get

$\rm \: (a + 7d) - (a + 3d) = 20$

$\rm \: a + 7d - a - 3d = 20$

$\rm \: 4d = 20$

$\rm\implies \:d = 5$

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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.