Question

the 6th term of an ap is -10 and its 10th term is -26. determine its 15th term

Answer 1

GivEn:-

• 6th term of an AP $\sf ( a_6 )$ = -10

• 10th term of an AP $\sf ( a_{10} )$ = -26

To find:-

• 15th term of AP $\sf ( a_{15} )$ is?

SoluTion:-

$\;\;\;\sf \star\;{\underline{\boxed{\sf{\pink{As\;per\;givEn\; QueStion\;:}}}}}\;\star$

✇ 6th term of an AP $\sf ( a_6 )$ = -10

$\dashrightarrow\sf a_6 = a + 5d$

$\dashrightarrow\sf -10 = a + 5d$ ----(1)

✇ 10th term of an AP $\sf ( a_{10} )$ = -26

$\dashrightarrow\sf a_{10} = a + 9d$

$\dashrightarrow\sf -26 = a + 9d$ ----(2)

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✩ ${\underline{\sf{\red{Substract\;eq(2)\;form\;eq(1)\;-}}}}$

⠀⠀⠀⠀⠀⠀⠀-10 = a + 5d

⠀⠀⠀⠀⠀⠀⠀-26 = a + 9d

⠀⠀⠀⠀⠀ ━━━━━━━━━━━

⠀⠀⠀⠀⠀ ⠀⠀16 = 0 - 4d

⠀⠀⠀⠀⠀ ━━━━━━━━━━━

$\therefore\;\sf We\;geT,$

$\dashrightarrow\sf 16 = -4d$

$\dashrightarrow\sf d = - \cancel{ \dfrac{16}{4}}$

$\dashrightarrow\sf \purple{d = - 4}$ ★

✩ ${\underline{\sf{\red{Now\;put\;value\;of\;d\;in\;eq(1)\;-}}}}$

$\dashrightarrow\sf -10 = a + 5(-4)$

$\dashrightarrow\sf -10 = a - 20$

$\dashrightarrow\sf \purple{a = 10}$ ★

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✩ ${\underline{\sf{\pink{Here,\;we\;have\;to\;find\;15^{th}\;term\;of\;AP\;-}}}}$

$\dashrightarrow\sf a_{15} = a + 14d$

✩ ${\underline{\sf{\red{Now\;put\;the\;givEn\;values,}}}}$

$\dashrightarrow\sf a_{15} = 10 + 14(-4)$

$\dashrightarrow\sf a_{15} = 10 - 56$

$\dashrightarrow\sf \purple{ a_{15} = - 46}$ ★

$\dag\;\sf \underline{Hence,\;15^{th}\;term\;of\;AP\;is\;- 46.}$

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Additional Information:-

$\begin{lgathered}\boxed{\begin{minipage}{10 em} \sf \displaystyle \bullet a_n=a + (n-1)d \\\\\\ \bullet S_n= \dfrac{n}{2} \left(a + a_n\right) \end{minipage}}\end{lgathered}$

Where,

• $\sf a_n$ is $\sf n^{th}$ terms of an AP.

• $\sf S_n$ is sum of n terms of an AP.

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