Question

The first and last terms of an ap are 1 and 11. If the sum of its terms is 36,then it's the number of terms will be​

Answer 1

Answer:

☮️Given that, the first and last term of an AP 1 and .

and the sum of its terms is 36

Let, the number of terms be n and common difference be d.

We know, the formula of n

the term of an A.P is

t n

=a+(n−1)d

And, the formula of term n− terms of an A.P is,

S n = 2 [2a+(n−1)d]

By the question

t n →1+(n−1)d=11⇒ d= n−1

10 ...(i)

S n → 2n

[2.1+(n−1)d]=36

⇒ 2+(n−1)d= n72

⇒ d= n(n−1)

72−2n ....(ii)

Comparing (i) and (ii) we get,

n(n−1)

10 = n(n−1)

72−2n

⇒ 10n=72−2n

⇒ 12n=72

⇒ n=6

Hence, the number of terms is 6

⛎hopes it helps you...

Answer 2

Given:

• $\sf{First\: term\;of\;AP,\;(a) = \textsf{ \textbf{1}}}$
• $\sf{Last\;term \;of \;AP,\;(l) = { \textsf{\textbf{11}}}}$
• $\sf{Sum\:of\;terms\;of\;AP,\: (s_{n}) ={ \textsf{ \textbf{36}}}}$

⠀⠀⠀

To find:

• The number of term

Solution:

▪︎For any Arithmetic Progression (AP), the sum of nth terms having last term is Given by :

$\begin{gathered} \blue\bigstar \underline{\boxed{{\sf{S_n = \dfrac{n}{2} \bigg\lgroup a + l \bigg\rgroup}}}}\\ \\\end{gathered}$

Where,

• n: no. of terms
• a: First Term
• l: Last Term

$\begin{gathered}\;\;\;\;{\underline{\frak{ \color{navy}{Substituting\;the\:given\;values\;in\;formula\;:}}}}\\\end{gathered}</p><p>⠀$

$\begin{gathered}:\implies\sf S_n = \dfrac{n}{2}\bigg\lgroup a + l\bigg\rgroup = 36\\ \\\end{gathered} \\ \\ \begin{gathered}:\implies\sf \dfrac{n}{2} \bigg\lgroup 1 + 11\bigg\rgroup = 36 \\ \\\end{gathered} \\ \\ \begin{gathered}:\implies\sf \dfrac{ \: n}{\cancel{\;2}} \times \: \cancel{12}= 36\\ \\\end{gathered} \\ \\ \begin{gathered}:\implies\sf 6n = 36\\ \\\end{gathered} \\ \\ \begin{gathered}:\implies\sf n = \cancel\dfrac{36}{6}\\ \\\end{gathered} \\ \\ \begin{gathered}:\implies\sf\underline{\boxed{{\frak{{n = 6}}}}}\pink\bigstar\\ \\\end{gathered} \\ \\ \begin{gathered}\therefore\:{\underline{\sf{Hence,\; number\:of\;terms\:of\;AP\;are\; {\textsf{\textbf{6}}}.}}}\\\end{gathered}$

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