20. If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term.

20. The given A.P. is 25,22,19, …

So, a=25

d=22 – 25= –3

Given that, sum of first n terms of the AP=116.

n [2 × 25+ (n – 1) (–3)]=116 × 2

n [50 – 3n+3]=232

n [53 – 3n]=232 .

53n – 3n²=232.

3n² – 53n+232.

Using quadratic formula, a=3, b= –53, c=232.

n = $\frac{53+5}{6}$ or $\frac{53-5}{6}$

= $\frac{58}{6}$ or $\frac{48}{6}$

= $\frac{29}{3}$ or 8.

As n $\in$ N, n=8.

? Last term=a+ (n – 1)d

=a+ (8 – 1)d

=25+7× (–3)

=25 – 21

=4.

<p><strong>20.&nbsp;</strong>The given A.P. is 25,22,19, …</p><p>So, a=25</p><p>d=22 – 25= –3</p><p>Given that, sum of first n terms of the AP=116.</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1729665775phpK7npSe_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1729665775phpK7npSe.jpeg" alt="" width="180" height="50"></picture></div></div><p>n [2 × 25+ (n – 1) (–3)]=116 × 2</p><p>n [50 – 3n+3]=232</p><p>n [53 – 3n]=232 .</p><p>53n – 3<em>n</em>²=232.</p><p>3<em>n</em>² – 53n+232.</p><p>Using quadratic formula, a=3, b= –53, c=232.</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1729665934php3AXICu_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1729665934php3AXICu_480x360.jpeg"4 alt="" width="220" height="149"></picture></div></div><p>n =&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>5</mn><mn>3</mn><mo>+</mo><mn>5</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></math></span>&nbsp;or&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>5</mn><mn>3</mn><mo>−</mo><mn>5</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></math></span></p><p>=&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>5</mn><mn>8</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></math></span>&nbsp;or&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>4</mn><mn>8</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></math></span></p><p>=&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>2</mn><mn>9</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math></span>&nbsp;or 8.</p><p>As n&nbsp;<span title="Click to copy mathml"><math><mrow><mo>∈</mo></mrow></math></span>&nbsp;N, n=8.</p><p>? Last term=a+ (n – 1)d</p><p>=a+ (8 – 1)d</p><p>=25+7× (–3)</p><p>=25 – 21</p><p>=4.</p>

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