Question

Find the sums given below:
7+10 1/2+14+.... +84​

Answer 1

✯Answer✯

☛Given:

• 7+ 10 1/2+ 14...84

☛To find:

• Sum of all terms of AP.

✯Concept to know✯

Before solving, we must analyze that what type of series or progression is this, so that we can easily find out the sum of all trms using formulas.

Here,

$\large{\mapsto{\sf{1st \: term(a1)= 7}}} \\ \\ \large{\mapsto{\sf{2nd \: term(a2) = 10 \frac{1}{2} }}} \\ \\ \large{\sf{\mapsto{3rd \: term = 14}}} \\ \\ \large{\sf{\implies{a2 - a1 = 10 \frac{1}{2} - 7 = 3 \frac{1}{2} }}} \\ \\ \large{ \sf{ \implies{a3 - a2 = 14 - 10 \frac{1}{2} = 3 \frac{1}{2} }}} \\ \\ \large{ \sf{ \red{ \star{ \: \: a2 - a1 = a3 - a2 = 3 \frac{1}{2} }}}}$

❋Hence, it is an Arithmetic progression, as the consecutive terms have common difference.

☛General formula for AP,

$\large{ \boxed</strong><strong>{</strong><strong>\</strong><strong>purple</strong><strong>{an = a + (n - 1)d}</strong><strong>}</strong><strong>}$

❋Note:

Symbols have their usual meanings

☛By using this formula,

$\large{ \mapsto{ \sf{84 = 7 + (n - 1)3 \frac{1}{2} }}} \\ \\ \large{ \mapsto{ \sf{84 = 7 + (n - 1) \frac{7}{2} }}} \\ \\ \large{ \mapsto{ \sf{77 = \frac{7}{2} (n - 1)}}} \\ \\ \large{ \sf{ \mapsto{ \cancel{77} \: \: 11 \times \frac{2}{ \cancel{7}} = n - 1}}} \\ \\ \large{ \sf{ \mapsto{ \boxed{n = 23}}}}$

☛Sum of n terms of an AP

$\large{ \sf{ \boxed</strong><strong>{</strong><strong>\</strong><strong>purple</strong><strong>{Sn = \frac{n}{2} \: \{ \: 2a + (n - 1)d \: \} }</strong><strong>}</strong><strong>}}$

❋Note:

Symbols have their usual meaning.

☛By using the formula,

$\large{ \sf{ \mapsto{S23 = \frac{23}{2} \{2 \times 7 + (23 - 1)3 \frac{1}{2} \} }}} \\ \\ \large{ \sf{ \mapsto{S23 = \frac{23}{2} \{ \: 14 +11\: \cancel{ 22} \times \frac{7}{ \cancel{2}} \} }}} \\ \\ \large{ \sf{ \mapsto{S23 = \frac{23}{2} \{14 + 77}}} \} \\ \\ \large{ \sf{ \mapsto{S23 = \frac{2093}{2} }}} \\ \\ \large{ \sf{ \mapsto{ \boxed{ \green{S23 = 1046.5}}}}}$

✯So final answer✯

$\large{ \boxed{ \bf{Sum \: of \: all \: terms = 1046.5}}}$

Answer 2

hope it may help you this answer is divided into 4 parts second answer is divided into two parts hope you understand which is the first part of second answer and second part of and also have a nice day please mark my answer as a bridge list and rate also ️