Question

An AP 5, 12, 19, ... has 50 terms. Find its last term. Hence, find the sum of its last 15 terms.

Answer 1

$\underline{\underline{\large{\mathfrak{Solution : }}}}$




$\underline{\mathsf{Given \: A.P. \longrightarrow 5 , 12 , 19 ,.............}} \\ \\ \mathsf{Here,} \\ \\ \mathsf{\implies First \: term (a) \: = \: 5 } \\ \\ \mathsf{\implies Common \: difference (d) \: = \: 12 \: - \: 5 \: = \: 7} \\ \\ \mathsf{\implies No. \: of \: terms \: = \: 50 } \\ \\ \mathsf{\implies Last \: term (l) \: = \: ?}$


$\underline{\textsf{Using Formula : }} \\ \\ \boxed{\mathsf{\implies l \: = \: a \: + \: ( n \: - \: 1)d}}$


$\mathsf{\implies l \: = \: 5 \: + \: ( 50 \: - \: 1)7} \\ \\ \mathsf{\implies l \: = \: 5 \: + \: 49 \: \times \: 7 } \\ \\ \mathsf{\implies l \: = \: 5 \: + \: 343 } \\ \\ \mathsf{\therefore \quad \: l \: = \: 348}$



$\underline{\textsf{Now, we have to find sum of last 15 terms.}} \\ \\ \mathsf{Here,} \\ \\ \mathsf{\implies First \: term ( a ) \: = \: ? } \\ \\ \mathsf{\implies Common \: difference (d) \: = \: 7 } \\ \\ \mathsf{\implies No. \: of \: terms (n) \: = \: 15 } \\ \\ \mathsf{\implies Last \: term (l) \: = \: 348} \\ \\ \mathsf{\implies Sum \: of \: 15 \: terms \: = \: ?}$



$\underline{\textsf{Using Formula : }} \\ \\ \boxed{\mathsf{\implies l \: = \: a \: + \: ( n \: - \: 1)d}}$



$\mathsf{\implies 348 \: = \: a \: + \: ( 15 \: - \: 1)7} \\ \\ \mathsf{\implies 348 \: = \: a \: + \: 14 \: \times \: 7 } \\ \\ \mathsf{\implies 348 \: = \: a \: + \: 98 } \\ \\ \mathsf{\implies a \: = \: 348 \: - \: 98 } \\ \\ \mathsf{\therefore \quad \: a \: = \: 250}$




$\textsf{Using Formula : } \\ \\ \boxed{\mathsf{\implies S_{n} \: = \: \dfrac{n}{2}(a \: + \: l)}} \\ \\ \mathsf{\implies S_{15} \: = \: \dfrac{15}{2}( 250 \: + \: 348 )} \\ \\ \mathsf{\implies S_{15} \: = \: \dfrac{15}{2} \: \times 598 } \\ \\ \mathsf{\implies S_{15} \: = \: 15 \: \times \: 299} \\ \\ \mathsf{\therefore \quad \: S_{15} \: = \: 4,485}$

Answer 2

$Here \: is \: the \: answer \: of \: your \: question$

$\textbf{According to question we have given that;}$

$\textbf{a = 5}$

$\textbf{n = 50}$

$\textbf{d = 12 - 5 = 7}$

$\textbf{an = ?}$

Formula used :-

$\textbf{an = a + (n - 1)d}$

$\textbf{an = 5 + (50 - 1) 7}$

$\textbf{an = 5 + (49 x 7)}$

$\textbf{an = 5 + 343}$

$\textbf{an = 348}$
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Now,

$\textbf{a = ?}$

$\textbf{S15 = ?}$

$\textbf{n = 15}$

$\textbf{an = 348}$

$\textbf{d = 7}$

Formula used :-

$\textbf{an = a + (n - 1) d}$

$\textbf{348 = a + (15 - 1) 7}$

$\textbf{348 = a + (14 x 7)}$

$\textbf{348 = a + 98}$

$\textbf{a = 250}$

Now, we have to S15. So, formula used is

$\textbf{Sn = n/2 [a + an]}$

$\textbf{S15 = 15/2 [250 + 348]}$

$\textbf{S15 = 15/2 (598)}$

$\textbf{S15 = 15 x 299}$

$\textbf{S15 = 4485}$
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